United States v. Martha Stewart and Peter Bacanovic

HALL, Circuit Judge. Defendants Martha Stewart and Peter Bacanovic appeal from the final judgments of conviction entered July 20, 2004 in the United States District Court for the Southern District of New York. Following trial before the Honorable Miriam Goldman Cedarbaum, the jury found Stewart and Bacanovic guilty of conspiracy, concealing material information from and making false statements to government officials, and obstructing an agency proceeding; the jury also found Bacanovic guilty of perjury. On March 17, 2004, this Court granted Stewart’s request for an expedited partial remand to permit the District Court to reconsider her sentence in light of United States v. Crosby, 397 F.3d 103 (2d Cir. 2005). On remand, the District Court decided not to modify the sentence that was imposed on July 16, 2004. United States v. Stewart, No. 03 CR 717(MGC), 2005 WL 831272 (S.D.N.Y. Apr.11, 2005). Bacanovic requests remand of his sentence under Crosby at this time. For the reasons set forth below, we conclude that none of the numerous grounds upon which Defendants challenge their convictions provides a basis to disturb the jury’s verdict and, therefore, we affirm the judgments of the District Court and remand the case solely for consider *280 ation of whether to modify Bacanovic’s sentence. BACKGROUND A. Procedural history Defendants Martha Stewart and Peter Bacanovie were charged in Superseding Indictment SI 03 Cr. 717 with offenses that arose from their communications to government investigators who were probing trading activity of ImClone Systems, Inc. (“ImClone”) stock on December 27, 2001, just ahead of the company’s public announcement that its lead pharmaceutical product would not receive government approval. Count One charged that Defendants conspired to obstruct justice, make false statements and commit perjury in violation of 18 U.S.C. § 371; Count Two charged Bacanovie with making false statements in violation of 18 U.S.C. § 1001(a)(1) and (2), and Counts Three and Four charged Stewart with the same offense; Count Five charged Bacanovie with making and using a false document in violation of 18 U.S.C. § 1001(a)(3); Count Six charged Bacanovie with perjury in violation of 18 U.S.C. § 1621; Counts Seven and Eight charged Bacanovie and Stewart, respectively, with obstructing an agency proceeding in violation of 18 U.S.C. § 1505; and Count Nine charged Stewart with securities fraud in violation of 15 U.S.C. §§ 78j(b), 78ff and 17 C.F.R. § 240.10b-5. 1 The trial lasted five weeks. At the close of evidence, pursuant to Fed.R.Crim.P. 29, the District Court granted Stewart’s motion for judgment of acquittal as to Count Nine. The jury deliberated for three days and returned a verdict convicting Stewart on specifications in Counts One, Three, Four and Eight and convicting Bacanovie on specifications in Counts One, Two, Six and Seven. The jury acquitted Stewart of one specification in Count Three and one specification in Count Four and acquitted Bacanovie of falsifying a worksheet document as charged in Count Five, as well as one specification in Count Two and several specifications in Count Six. The District Court denied Defendants’ post-trial motions for a new trial. On July 16, 2004, the District Court sentenced each Defendant to five months’ incarceration to be followed by a two-year period of supervised release, five months of which were to be served in home confinement. Stewart and Bacanovie were ordered to pay fines of $30,000 and $4,000, respectively, as well as a mandatory $400 special assessment. Anticipating a decision from the Supreme Court addressing the United States Sentencing Guidelines, the District Court stayed execution of the sentences pending appeal. The stays were subsequently vacated and amended judgments of conviction were entered as to Stewart on September 22, 2004, and as to Bacanovie on December 29, 2004. In this appeal, Stewart, who had already served the period of incarceration, requested immediate remand of the supervised release portion of the judgment, pursuant to Crosby, to give the District Court an opportunity to consider whether to modify the sentence in light of the Supreme Court’s intervening decision in United States v. Booker, 543 U.S. 220, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005). This Court granted Stewart’s application and, on remand, the District Court declined to modify the original sentence, concluding that it would have imposed the same sentence even if the Sentencing Guidelines *281 had not been mandatory at the time of sentencing. See Stewart, 2005 WL 831272 at *1. Bacanovic, who completed the incarceration portion of his sentence in June 2005, now requests that his sentence be remanded to the District Court for consideration of whether to modify under Crosby. His application is granted. B. The trial At trial, the Government sought to prove that Stewart and Bacanovic conspired and acted to mislead the ImClone investigation in order to deflect attention from the fact that, on December 27, 2001, Stewart sold shares of ImClone from her personal account at Merrill Lynch & Co. (“Merrill Lynch”) after she learned from Bacanovic, her broker, that ImClone’s CEO, Samuel Waksal, was attempting to sell all of his own shares in the company. In connection with the investigation, Stewart was interviewed twice, on February 4, 2002 and April 10, 2002, by the Securities and Exchange Commission (“SEC”), the Federal Bureau of Investigation (“FBI”) and members of the United States Attorney’s Office for the Southern District of New York (the “U.S. Attorney”). Those agencies interviewed Bacanovic on January 7, 2002, and he testified before the SEC on February 13, 2002. At trial, the Government offered the testimony of SEC attorney Helene Glotzer and FBI agent Catherine Farmer, who attended each of the Defendants’ interviews, to inform the jury of what Stewart and Bacanovic said — and did not say— about Stewart’s ImClone investment, its liquidation, and the Defendants’ communications regarding those matters on and after December 27, 2001. In addition, the jury heard a tape recording of Bacanovic’s SEC testimony. To demonstrate that the story Defendants told to investigators was a cover-up of the events of December 27th, the Government called a number of witnesses to testify about their recollections of what happened that day and in the following months. Various portions of the testimony of those witnesses were corroborated by phone records, copies of emails and phone message logs. Among those who testified in the Government’s case-in-chief were Stewart’s assistant Ann Armstrong, Stewart’s friend Mariana Pasternak, Sam Waksal’s assistant Emily Perret, Bacanovic’s assistant Douglas Faneuil, who appeared pursuant to a cooperation agreement, and Merrill Lynch compliance personnel. Both the prosecution and the defense offered testimony from securities analysts on the volume and price movement of ImClone on and around December 27th. The jury heard opinion testimony regarding Bacanovic’s allegedly altered worksheet from the Defendant’s ink expert and from the Government’s ink expert, Lawrence F. Stewart, whose testimony later created a separate controversy (see Point II.A, infra). Bacanovic called one of his clients to testify about his character and expertise as well as certain trading practices and preferences. He also offered the testimony of Faneuil’s former attorney and Stewart’s business manager to rebut parts of Faneuil’s testimony. Stewart called one witness, an attorney who attended and took notes at the February 4th interview. The following story unfolded at trial. In the fall of 2001, Stewart was the CEO of Martha Stewart Living Omnimedia, Inc. (“MSLO”), and Bacanovic was a stock broker at Merrill Lynch. Among Bacanovic’s clients were Stewart, Samuel Waksal, who was then the CEO of ImClone, and Waksal’s daughter Aliza. At that time, ImClone had great expectations for its lead product, the cancer-treating drug Erbitux. The biotechnology company was anticipating *282 that the. Food and Drug Administration (“FDA”) would approve its application for the drug by early 2002. With the prospect of commercialization on the horizon, Bristol-Myers Squibb made a tender offer to purchase ■ 20 percent of ImClone’s outstanding shares at a price of $70 per share and agreed to fund ImClone’s continued development of Erbitux while undertaking responsibility for sales and marketing following FDA approval. In October 2001, the MSLO pension fund held 51,800 shares of ImClone, apparently acquired over the course of the previous decade, and Stewart owned an additional 5,000 shares personally. All of the shares in the MSLO pension fund and those held by Stewart individually were tendered in response to Bristol-Myers’ offer, and all but 3,928 of the shares in Stewart’s personal account were sold. Stewart’s remaining ImClone stock represented approximately 10 percent of her total Merrill Lynch portfolio in November 2001. • 1. The events of December 27th, as told by Faneuil, Armstrong, Pasternak and Perret During the last week of December 2001 and the first week of January 2002, Bacanovic was vacationing in Florida. Douglas Faneuil, a Merrill Lynch client associate in his mid-twenties who had been working as Bacanovic’s assistant for about six months, was responsible for covering Bacanovic’s desk when he was away. Between 9:00 and 10:00 on the morning of December 27th, Faneuil received several phone calls on Bacanovic’s line from Sam Waksal’s daughters, Aliza Waksal and Elana Waksal Posner, and from Waksal’s accountant, Alan Goldberg. Prior to the opening of the market, Aliza Waksal placed a market order to sell all of her approximately 40,-000 ImClone shares, which Faneuil executed. Shortly thereafter, Faneuil received a call from Alan Goldberg directing him to sell all of Sam Waksal’s ImClone shares immediately. Faneuil advised Goldberg that he was unable to do so because of SEC rules restricting Waksal’s trading of his company’s shares. Goldberg informed Faneuil that, pursuant to facsimile instructions from Waksal that were dated December 26th (but not received at Merrill Lynch until December 27th), he was to transfer Waksal’s entire holding of approximately 80,000 ImClone shares to the account of his daughter Aliza and sell it from there. Faneuil confirmed — with both Bacanovic and Merrill Lynch compliance personnel — his understanding that the .proposed transaction was not permissible. Faneuil spoke with Goldberg five or six times by phone that day, and the transfer between accounts took place the following day, December 28th. Faneuil also received two calls that morning from Elana Waksal Posner regarding the sale of her ImClone shares; she expressed disappointment that the price was “already going down.” Faneuil kept Bacanovic apprised of the details of the calls from the Waksal family by telephone. In the midst of one of their conversations, Faneuil heard Bacanovic say suddenly, “Oh, my God, get Martha [Stewart] on the phone.” With Bacanovic on the line, Faneuil placed a call to Stewart’s New York office. Bacanovic left a message after being told by Stewart’s assistant, Ann Armstrong, that Stewart was traveling. Armstrong logged the message as follows: “Peter Bacanovic thinks ImClone is going to start trading downward.” In a follow-up call, Bacanovic instructed Faneuil “to tell [Stewart] what’s going on” when she returned the call. Faneuil recalled that Bacanovic expressly confirmed that Faneuil was to advise Stewart of Waksal’s efforts to sell his ImClone stock, a communication that Faneuil knew would *283 violate Merrill Lynch’s client confidentiality policy. At 1:18 p.m. Bacanovie sent an email to Faneuil inquiring as to whether any news had “come out” regarding ImClone; Faneuil responded that there had been nothing yet. Shortly thereafter, Stewart — who was en route to Mexico for a vacation with her friend Mariana Pasternak — called her office and was informed by Armstrong of Bacanovic’s message. Armstrong transferred the call to Bacanovic’s office, where Faneuil answered, “Merrill Lynch, Peter Bacanovic’s office.” It is unclear whether Faneuil identified himself to Stewart. Faneuil told Stewart that, although there had been no news release from ImClone, Bacanovic thought she would “like to act on the information that Sam Waksal was trying to sell all of his shares,” at least those he held in Merrill Lynch accounts. Stewart asked Faneuil for a price quote and directed him to sell all of the ImClone shares that remained in her portfolio. Stewart then placed a call to Sam Waksal, reaching his secretary Emily Perret. Stewart asked Perret if she knew what was going on with ImClone and told Perret to find Waksal. Perret informed Stewart that Waksal was not in the office, and she noted the call on Waksal’s messages sheet for December 27, 2001 as follows: 1:43 Martha Stewart something is going on with ImClone and she wants to know what She is on her way to Mexico and she is staying at Los Vantanos [sic]. Back in New York, Stewart’s ImClone sell order was executed at an average price of $58.43 per share, yielding proceeds of approximately $230,000. Pursuant to Stewart’s instructions, Faneuil sent an email to her personal account confirming the trade. Faneuil also informed Baeanovic that, after hearing that Waksal was trying to sell all of his ImClone shares, Stewart sold her own shares. Several days later, while Stewart and Mariana Pasternak were visiting in Mexico, their discussion turned to the New Year’s plans of various friends, and Pasternak inquired about Sam Waksal. Among other things, Stewart told Pasternak about the decline in ImClone’s stock price, Waksal’s efforts to sell his holdings, and her own sale. Pasternak later testified that she was either told by Stewart or was left with the impression that it was “nice to have brokers who tell you those things,” and she acknowledged knowing that Bacanovie was Stewart’s broker. 2. The ImClone investigation and events after December 27th, as told by Faneuil, Armstrong, Glotzer, Farmer and Merrill Lynch compliance personnel a. the investigation begins On December 28, 2001, the day following Stewart’s ImClone sale, ImClone announced that the FDA had rejected its application for Erbitux approval. When the market next opened, on December 31, 2001, ImClone’s stock price had dropped approximately 18 percent to $45.39 per share. Prompted by these developments, Merrill Lynch compliance personnel reviewed ImClone trade data preceding the announcement. Upon discovery of the sales by the Waksals and Stewart — all clients of Bacanovie — the matter was referred to Merrill Lynch senior management, who directed further inquiry. Merrill Lynch compliance personnel contacted Faneuil and Bacanovie on December 31st with questions about the ImClone trades. Bacanovie, who was still out of town, told Merrill Lynch administrative manager Julia Monaghan that Faneuil handled the trades. He also said that Stewart’s sale was part of her planned year-end tax loss selling. Monaghan then *284 sought out Faneuil. Immediately after speaking with Monaghan, Faneuil called Baeanovic to confirm the accuracy of an answer he had given to Monaghan. Bacanovic, as though coaching Faneuil, repeatedly stated that Stewart’s trade was made pursuant to a pre-existing plan for year-end tax loss selling to offset gains in other investments. Faneuil knew, however, that the timing of the ImClone sale and its gain to Stewart were inconsistent with a tax loss strategy. b. Faneuil’s January 3rd interview Based on the results of its internal review, Merrill Lynch referred the ImClone matter to the SEC. The SEC launched an investigation, as did the FBI and the U.S. Attorney. On January 3, 2002, the SEC interviewed Faneuil by phone, focusing the inquiry on the Waksal family’s trading on December 27th. With regard to Stewart’s trade, Faneuil explained only that Stewart had called, requested a quote and decided to sell. He then reported the substance of the SEC interview to Baeanovic, using a borrowed cell phone to avoid any possibility that the call would be preserved on a Merrill Lynch taped line. On the following Monday, Faneuil approached Merrill Lynch about retaining outside counsel for him. Soon after the January 3rd SEC interview, Faneuil received a call from Stewart’s business manager, Heidi DeLuca, complaining that the ImClone sale generated a gain that compromised Stewart’s tax loss selling plan and created a tax liability. Faneuil reported DeLuea’s call to Baeanovic. Baeanovic then gave Faneuil a different explanation for the trade, insisting that Stewart sold the ImClone stock on December 27th pursuant to a preexisting agreement to sell if the price dropped to $60 per share, although no such order had been entered into Merrill Lynch’s computer system. Called as a defense witness at trial, DeLuca confirmed the existence of that stop-loss order. She asserted, however, that the conversation with Faneuil regarding the ImClone gain took place in February, rather than early January as Faneuil had recalled. c. Bacanovic’s January 7th interview Upon returning to the office from vacation on January 7th, Baeanovic was questioned again by Merrill Lynch’s Monaghan about Stewart’s trade. Baeanovic told her that earlier in December, while he and Stewart were reviewing Stewart’s portfolio and tax loss strategy, they decided to sell ImClone if the price dropped to or below $60 per share. He recounted the $60 per share stop-loss order story to the SEC in a telephone interview later that day, explaining that on December 27th he advised Stewart that ImClone had dropped below $60, and she told him to sell it. After speaking to the SEC, Baeanovic took Faneuil out for coffee and a talk. Baeanovic explained Stewart’s integral role in advancing his career and stressed his loyalty to her. Faneuil brought up the events of December 27th and reminded Baeanovic that he knew what really transpired, at which point Baeanovic asserted that Faneuil did not know what was going on that day and admonished Faneuil for being selfish. When Faneuil returned from a week’s vacation in mid-January, Baeanovic told him that he had met recently with Stewart and discussed the events of December 27th with her. Stewart’s calendar, which Armstrong maintained, reflected a breakfast meeting with Baeanovic on January 16th. According to Faneuil, Baeanovic said to him, “Everyone’s telling the same story. This was a $60 stop-loss order. That was the reason for her sale. We’re all on the *285 same page, and it’s the truth. It’s the true story. Everyone’s telling the same story.” d. Stewart’s February 4th interview On or around January 22, 2002, Stewart consulted with counsel from Wachtell, Lipton, Rosen & Katz, whose records reflect that work on “MS/ImClone matters” would be undertaken. Several days thereafter, Stewart’s attorneys were contacted by the offices of the U.S. Attorney and the FBI, who asked to speak with her. Over the next week, Stewart met with and spoke by phone to her attorneys several times. At the end of one such call on January 31st, Stewart asked Armstrong to send to the law firm copies of messages that she had received during the period from December 26, 2001 through January 7, 2002. Stewart then asked Armstrong to show her the messages, including the entry for the December 27th message from Bacanovic. Upon reading the message, Stewart took the computer mouse from Armstrong and deleted and typed over a portion of the text so that what initially read, “Peter Bacanovic thinks Imclone is going to start trading downward” was revised to read, “Peter Bacanovic re imclone [sic].” Immediately thereafter, Stewart told Armstrong to restore the message to the original, and Armstrong did so. On February 4, 2002, in response to the SEC’s request for an interview, Stewart met at the offices of the U.S. Attorney with two SEC enforcement attorneys, an Assistant United States Attorney and an FBI agent, all of whom were investigating the December 27th ImClone trading. Stewart told the investigators that in the fall of 2001, shortly after selling a portion of her ImClone stock to Bristol-Myers, she decided with Bacanovic to sell the remainder if the price fell to $60 per share. Stewart recounted receiving a message to call Bacanovic while she was en route to Mexico. Stewart said that she called Bacanovic, who advised her that ImClone shares were trading below $60, and directed him to sell all of her shares. She explained to the investigators that she wanted to take care of the matter at that time rather than be bothered during her vacation. She stated that she spoke with Bacanovic on December 27th, and she denied speaking with his assistant. Stewart added that during the call she and Bacanovic also discussed her company’s stock as well as K-Mart. Although Stewart had reviewed the message log reflecting the call from Bacanovic only four days earlier at Armstrong’s desk, Stewart denied in the interview that she knew whether there was a written record of Bacanovic’s message but agreed to check into the matter. Stewart also denied that she had discussed ImClone with Bacanovic during the week leading up to December 27th and said that her several discussions with him since that time had been limited to matters in the “public arena.” Although Stewart told investigators that Bacanovic had informed her that the SEC was questioning Merrill Lynch about the December 27th trading, Stewart also said that Bacanovic did not tell her whether he had been questioned or whether any questions involved her. Stewart said nothing at the interview about being aware of the Waksals’ intentions to sell their ImClone shares on December 27th. e. Bacanovic’s February 13th testimony Bacanovic gave sworn testimony to the SEC on February 13, 2002, responding to questions about ImClone trading in the Waksal family accounts that he managed and about Stewart’s holdings, including ImClone and her decision to sell on December 27th. He denied telling Stewart on December 27th of Waksal’s efforts to *286 sell his ImClone stock and stated that he would never discuss one client’s transactions with another. Describing the events of December 27th, Bacanovic told investigators that Faneuil’s phone calls to him regarding the Waksal family prompted him to remember Stewart’s decision to sell her ImClone shares if the price dropped to $60 per share. Bacanovic commented that Stewart never thought that would happen. Bacanovic testified that Stewart made up her mind to sell the declining ImClone shares, which he believed she held out of loyalty to Waksal, at $60 per share during a “comprehensive portfolio review” with him on December 20th. He referred to a contemporaneous worksheet, which was later produced to the SEC, reflecting that and other decisions. Bacanovic explained that no order was entered into Merrill Lynch’s computer system to trigger the sale of ImClone because Stewart, like most of his clients, eschewed automatic execution in favor of having him track a stock and give notice if and when it reached the target price. Bacanovic told investigators that on December 27th he left a message with Stewart’s assistant, advising her of ImClone’s price and asking that Stewart “[c]all my office.” He stated that Faneuil later reported to him that Stewart called Bacanovic’s line on the 27th and directed him to sell her shares. Bacanovic denied speaking with Stewart on that day. He also represented that in conversations since December 27th he and Stewart had discussed ImClone in general terms and that he had informed her of an internal Merrill Lynch review. He denied speaking with Stewart about her own ImClone trades, the government investigation, or the fact that he had been questioned about the events of December 27th. f. Faneuil’s March 7th interview On March 7, 2002, several weeks after Bacanovic’s SEC testimony, Faneuil was interviewed by representatives from the SEC, the FBI and the U.S. Attorney’s Office. He stated, as he had previously, that on December 27th, Stewart called for a quote and decided to sell. He did not mention to the investigators that during the call he informed Stewart of Waksal’s efforts to sell, nor did he say that Bacanovic had told him to do so. When Faneuil reported this to Bacanovic, Bacanovic replied, “good.” g. Stewart’s April 10th interview After receiving a copy of Sam Waksal’s phone log, which reflected a call from Stewart on the morning of December 27th, investigators requested another meeting with Stewart, and she agreed to speak with them. During that April 10, 2002 interview, Stewart stated that she had no recollection of having been told on December 27th that any of the Waksals were selling their ImClone stock. Stewart was asked about Bacanovic’s December 27th phone message, which Armstrong had entered in the log as “Peter Bacanovic thinks ImClone is going to start trading downward.” Stewart stated that she did not recall that Armstrong told her about the content of Bacanovic’s message. Rather she recalled being told that Bacanovic called and wanted to speak with her before the end of the day. On April 10th, the investigators also questioned Stewart about the message 'she left with Sam Waksal’s secretary on the morning of December 27th, asking what was “going on with ImClone.” Stewart told investigators that she called Waksal on the 27th for the purpose of seeing how he was and making sure everything was okay. She also stated that when she was interviewed two months earlier on February 4th, she did not recall having made the call to Waksal. *287 h. Faneuil’s decision to contact the government During the period between February and May 2002, Bacanovic talked to Faneuil about the investigation approximately five times, each time reiterating to Faneuil that he had spoken with Stewart and that everyone was “on the same page” and telling the same $60 stop-loss story, which was the “truth.” During one such conversation in April or May, Faneuil reminded Bacanovic that it was Faneuil who spoke with Stewart on December 27th and that he knew what actually had been said. At that point, according to Faneuil, Bacanovic said, “don’t even say that, just don’t even say that.” Faneuil admitted at trial, however, that Bacanovic never explicitly told him to lie. A month or two later, in June 2002, although he had not been served with a subpoena, Faneuil admitted to Merrill Lynch and to the government investigators that he had lied twice to the SEC about the content of his December 27th phone conversation with Stewart. Faneuil testified at trial that the lies and subsequent cover-up became too much to bear. Faneuil entered into a cooperation agreement with the Government, pleading guilty to the misdemeanor charge of receiving money or things of value as a consideration for not informing against a violation of the law in violation of 18 U.S.C. § 873. He agreed with the SEC to a lifetime exclusion from work in the securities industry. 3. Stewart’s June 2002 public statements In June 2002, media reports stated that a congressional investigation into the FDA’s denial of the Erbitux application had revealed the December 27th ImClone transactions by Waksal and Stewart. Thereafter, the value of MSLO shares declined, which resulted in a corresponding decrease in Stewart’s net worth. Stewart issued two public statements, one in a press release and one addressed to a conference of securities analysts and advisors. Stewart explained that her sale had been triggered by the pre-existing decision to sell ImClone if and when it reached $60 per share and not by information that was unavailable to the public. She also represented that she had cooperated fully with government investigators. MSLO shares enjoyed-a modest rebound. C. Judgment of acquittal on Count Nine Following the close of evidence and before summations, the District Court granted Stewart’s motion for judgment of acquittal, pursuant to Fed.R.Crim.P. 29, on Count Nine, which charged Stewart with defrauding MSLO investors by making false public statements in June 2002 (see Point B.3, supra) for the purpose of deceiving investors and thwarting the decline in the value of her own MSLO stock. See United States v. Stewart, 305 F.Supp.2d 368 (S.D.N.Y.2004). The District Court found that the evidence was insufficient to establish that Stewart intended to defraud investors when she issued the statements, holding that “a reasonable juror could not, without resorting to speculation and surmise, find beyond a reasonable doubt that Stewart’s purpose was to influence the market in MSLO securities.” Id. at 376. D. The verdict The jury began deliberating on March 3, 2004. The District Court provided the jurors with (i) a redacted version of the Superseding Indictment, identifying the charged acts and omissions as well as the statutory allegations and (ii) a verdict form listing each specification in the false statement and perjury counts with instructions that for any count of conviction a check *288 mark be placed next to any specification found. On the third day of deliberations, March 5th, the jury returned a verdict acquitting Defendants of certain charges and finding them guilty on others. The District Court summarized the jury’s specific findings as follows: The jury convicted Stewart of making false statements to investigators during her February 4 interview, in violation of 18 § U.S.C. 1001. The jury found Stewart guilty of making the following false statements, each of which was a specification in Count Three of the Indictment. Stewart told the Government investigators that she spoke to Bacanovic on December 27 and instructed him to sell her ImClone shares after he informed her that ImClone was trading below $60 per share. Stewart also stated that during the same telephone call, she and Bacanovic discussed the performance of the stock of her own company, Martha Stewart Living Omnimedia (“MSLO”), and discussed K-Mart. She told investigators that she had decided to sell her ImClone shares at that time because she did not want to be bothered during her vacation. Stewart stated that she did not know if there was any record of a telephone message left by Bacanovic on December 27 in her assistant’s message log. She also said that since December 28, she had only spoken with Bacanovic once regarding ImClone, and they had only discussed matters in the public arena. Finally, Stewart told investigators that since December 28, Bacanovic had told her that Merrill Lynch had been questioned by the SEC regarding ImClone, but that he did not tell her that he had been questioned by the SEC or that he had been questioned about her account. The jury acquitted Stewart of one specification charged in Count Three: her statement that she and Bacanovic had agreed, at a time when ImClone was trading at $74 per share, that she would sell her shares when ImClone started trading at $60 per share. The jury found Stewart guilty of making the following false statements to investigators during her April 10 interview. Each of these statements was a specification in Count Four of the Indictment. Stewart said that she did not recall if she and Bacanovic had spoken about Waksal on December 27 and that she did not recall being informed that any of the Waksals were selling their ImClone stock. Stewart also reiterated that she spoke to Bacanovic on December 27, that he told her the price of ImClone shares, and that he suggested that she sell her holdings. The jury did not find Stewart guilty of one false statement specification charged in Count Four: her statement that sometime in November or December of 2001, after she sold ImClone shares held in the Martha Stewart Defined Pension Trust, she and .Bacanovic decided she would sell her remaining ImClone shares when they started trading at $60 per share. The jury found Bacanovic guilty of making one false statement during his January 7 interview with the SEC, in violation of 18 U.S.C. § 1001. This was a specification in Count Two of the Indictment, which charged Bacanovic with falsely stating that he had spoken to Stewart on December 27, that he told Stewart during that conversation that ImClone’s share price had dropped, and that Stewart had instructed him to sell her shares. The jury found Bacanovic not guilty of the other false statement charged in *289 Count Two: his statement that on December 20, 2001, he had a conversation with Stewart in which she decided to sell her ImClone stock at $60 per share. The jury also convicted Bacanovic of perjury in violation of 18 U.S.C. § 1621, for one statement he made during his February 13 testimony before the SEC. Perjury was the charge in Count Six of the Indictment. Bacanovic stated that on the morning of December 27, he had left a message for Stewart with her assistant, Ann Armstrong. He said that the message requested that Stewart return his call, and advised her of the price at which ImClone was then trading. The jury acquitted Bacanovic of five other perjury specifications charged in Count Six. These specifications related to conversations Bacanovic had had with Stewart subsequent to her December 27 trade, the circumstances of her decision on December 20 to sell ImClone at $60 per share, and a worksheet he had used during their December 20 conversation. The jury acquitted Bacanovic of a charge of making and using a false document, which was charged as a violation of 18 U.S.C. § 1001 in Count Five of the Indictment. This count was based on a worksheet that Bacanovic gave the SEC in the course of their investigation. Bacanovic claimed that he had used the worksheet during his December 20 conversation with Stewart. The worksheet listed Stewart’s holdings and contained numerous handwritten notations in blue ink. The bullet point before ImClone’s entry on the worksheet was circled in blue ink, as were the bullet points preceding several other entries on the page. Beside ImClone’s name was a notation, “@60,” also in blue ink.. The “@60” notation was the basis of the charge. The jury also convicted defendants of conspiracy [in violation of 18 U.S.C. § 371] and obstruction of an agency proceeding in violation of 18 U.S.C. § 1505. With respect to the conspiracy charge, the jury found that the defendants conspired to carry out all three objects of the conspiracy: making false statements, perjury, and obstruction of an agency proceeding. United, States v. Stewart, 323 F.Supp.2d 606, 609-10 (S.D.N.Y.2004) (footnote identifying appendices omitted). We agree with the District Court’s overview of the jury’s findings: [F]irst, that the jury found that the Government had not proved, beyond a reasonable doubt that defendants had fabricated the $60 agreement; and second, that the jury found beyond a reasonable doubt that Stewart and Bacanovic agreed to lie and did lie to Government investigators to conceal the fact that when Stewart sold her ImClone stock on December 27, 2001, she had been tipped by Bacanovic’s assistant that the CEO of ImClone was trying to sell his ImClone shares held at Merrill Lynch. Id. at 614. DISCUSSION Stewart and Bacanovic both argue that we must reverse their convictions, order a new trial, or remand for an evidentiary hearing because the trial was tainted by Sixth Amendment violations, prosecutorial misconduct, juror misconduct, extraneous influences on the jury, and erroneous evidentiary rulings and jury instructions. Bacanovic also advances additional arguments pertaining solely to his conviction. None of Defendants’ appellate arguments persuades us to grant the requested relief. *290 L Confrontation Clause The Government called SEC enforcement attorney Helene Glotzer and FBI agent Catherine Farmer to introduce the statements that Stewart made to investigators in interviews on February 4th and April 10th and the statements that Bacanovie made in the January 7th interview. The jury heard the tape recording of Baeanovic’s February 13th testimony answering questions about Stewart, her ImClone trade and their communications regarding the trade and the investigation. Each Defendant now complains that admitting into evidence certain portions of those statements for the truth of the matters asserted by one of them to prove facts used to convict the other violated their respective rights under the Confrontation Clause 2 as articulated by the Supreme Court in Crawford v. Washington, 541 U.S. 36, 124 S.Ct. 1354, 158 L.Ed.2d 177 (2004). Those errors; they assert, mandate reversal of the judgments of conviction. Crawford, which was issued just days after the verdict in this case, announced a per se bar on the admission of a class of out-of-court statements, denominated “testimonial,” against an accused who had no prior opportunity to cross-examine the declarant. Crawford, 541 U.S. at 68-69, 124 S.Ct. 1354. While declining to “spell out a comprehensive definition of ‘testimonial,’ ” id. at 68, 124 S.Ct. 1354, the Supreme Court identified certain examples at the “core” of the definition, id. at 51-52, 124 S.Ct. 1354. “[T]he types of statements cited by the Court as testimonial' share certain characteristics; all involve a declarant’s knowing responses to structured questioning in an investigative environment or a courtroom setting where the declarant would reasonably expect that his or her responses might be used in future judicial proceedings.” United States v. Saget, 377 F.3d. 223, 228 (2d Cir.2004). Stewart contends that Bacanovic’s statements in interviews conducted by representatives of the SEC, U.S. Attorney’s Office, and FBI, as well as his taped sworn testimony before the SEC, are the type of statements that Crawford bars in the absence of an opportunity for cross-examination. Bacanovic asserts the same argument with respect to statements made by Stewart during the course of the February 4th and April 10th interviews. Defendants further contend that there is no ground on which to except the challenged statements from this requirement because they are “testimonial” under Crawford and because the statements of each of them were offered for the truth of the matter asserted as probative of the other’s guilt. Neither Defendant objected at trial to admission of these challenged statements. Accordingly, we review their Sixth Amendment Crawford claims for plain error. See United States v. Bruno, 383 F.3d 65, 78 (2d Cir.2004). We find none. A. Standard of review Unpreserved Confrontation Clause claims are reviewed for plain error. Id. Under Fed.R.Crim.P. 52(b), an error that was not raised at trial may not be considered unless it is (1) an “error,” meaning an unwaived deviation from a legal rule, that is (2) “plain,” which means clear and obvious under the law at the time of appellate review, and (3) “affect[s] substantial rights” by influencing the outcome of the trial. Johnson v. United States, 520 U.S. 461, 466-67, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997) (internal quo *291 tation marks omitted; alteration in original); United States v. Olano, 507 U.S. 725, 732-35, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993) (internal quotation marks omitted). Even if such an error is noted, the discretion to correct it under Rule 52(b) is to be exercised only if the error “seriously affect[s] the fairness, integrity, or public reputation of judicial proceedings.” Johnson, 520 U.S. at 467, 117 S.Ct. 1544 (internal quotation marks omitted; alteration in original); Olano, 507 U.S. at 735-37, 113 S.Ct. 1770. B. Analysis 1. The District Court did not err in admitting the challenged statements Turning to the first prong of plain error analysis, we find that there was no error in the admission of each of these co-defendant’s statements against the other. Although the statements at issue, having been made during interviews with government officials in the course of an investigation, do have characteristics of Crawford’s “core class of ‘testimonial’ statements,” Crawford, 541 U.S. at 51-52, 124 S.Ct. 1354; Saget, 377 F.3d at 229, in the context of the crimes for which Defendants were convicted, the challenged statements are part and parcel of co-conspirators’ statements made in the course of and in furtherance of Defendants’ conspiratorial plan to mislead investigators. See Fed. R.Evid. 801(d)(2)(E). But for the fact that they were made in response to inquiries by government agents during formal investigations and thus may have an aura of “testimony,” there is no question that the contents of all such statements of one member of the conspiracy would be admissible against other members. See Crawford, 541 U.S. at 56, 124 S.Ct. 1354. To admit a statement as nonhearsay under Rule 801(d)(2)(E), there must be a preponderance of evidence independent of the proffered statements demonstrating that there was a conspiracy, that the declarant and the person against whom the statement is offered belonged to the conspiracy, and that the statements were made during the course of and in furtherance of the conspiracy. See, e.g., United States v. Alameh, 341 F.3d 167, 176-77 (2d Cir. 2003). Without considering the contents of the statements themselves, the record here contains ample evidence of a conspiracy among Stewart, Bacanovie and Faneuil to cover up what they said and did on and after December 27th by responding to inquiries instead of asserting their Fifth Amendment rights not to incriminate themselves, for example, and by giving false and misleading responses to certain of the investigators’ questions while shrouding those responses with an aura of accurate detail. Here, Defendants do not have the temerity to argue that somehow Crawford precludes the government’s proof of the Defendants’ false portions of their statements because they were provided in a testimonial setting. Crawford expressly confirmed that the categorical exclusion of out-of-court statements that were not subject to contemporaneous cross-examination does not extend to evidence offered for purposes other than to establish the truth of the matter asserted. See Crawford, 541 U.S. at 58-59, 124 S.Ct. 1354 & n. 9 (“The [Confrontation] Clause also does not bar the use of testimonial statements for purposes other than establishing the truth of the matter asserted.”) (citing Tennessee v. Street, 471 U.S. 409, 414,105 S.Ct. 2078, 85 L.Ed.2d 425 (1985)); see also United States v. Logan, 419 F.3d 172, 177-78 (2d Cir.2005) (same). Defendants object that certain truthful portions of their statements made during the course of the agreed-upon obstruction must be excluded because they are “testimonial.” On the *292 facts of this case, where the object of the conspiracy is to obstruct an investigation that is engaged in obtaining those testimonial statements of the conspirators, that objection must fail. Understanding the reasons for that conclusion requires further consideration of Crawford. Crawford set forth two propositions that are clear when considered separately but problematic when they arise in the same case. First, the Supreme Court stated that statements in furtherance of a conspiracy were generally not testimonial and were exceptions to the hearsay rule that encountered no Confrontation Clause obstacle. See Crawford, 491 U.S. at 56, 109 S.Ct. 2273. Second, the Court stated that out-of-court testimonial statements, if not subjected to cross-examination, did encounter a Confrontation Clause obstacle. See id. at 50-56, 109 S.Ct. 2273. Crawford had no occasion to consider the situation we face: statements that are both in furtherance of a conspiracy and testimonial. We need not attempt to resolve broadly the tension between the “in furtherance” rule allowing uncross-examined statements and the “testimonial” rule prohibiting them because in the pending case that tension arises in a special context, and the context itself points toward a resolution of the tension. 3 Here the truthful portions of the testimonial statements were made in furtheranee of a conspiracy to obstruct justice. The essence of such a conspiracy necessarily contemplates that the conspirators would provide false information to government agencies during the course of their investigation and during interrogations that would produce testimonial statements of one or the other of them. It defies logic, human experience and even imagination to believe that a conspirator bent on impeding an investigation by providing false information to investigators would lace the totality of that presentation with falsehoods on every subject of inquiry. To do so would be to alert the investigators immediately that the conspirator is not to be believed, and the effort to obstruct would fail from the outset. As noted, the admissibility of such totally false statements, made in the course and in furtherance of the conspiracy, suffers no Sixth Amendment bar under Crawford. The truthful portions of statements in furtherance of the conspiracy, albeit spoken in a testimonial setting, are intended to make the false portions believable and the obstruction effective. Thus, the truthful portions are offered, not for the narrow purpose of proving merely the truth of those portions, but for the far more significant purpose of showing each conspirator’s attempt to lend credence to the entire testimonial presentation and thereby obstruct justice. It would be unacceptably *293 ironic to permit the truthfulness of a portion of a testimonial presentation to provide a basis for keeping from a jury a conspirator’s attempt to use that truthful portion to obstruct law enforcement officers in their effort to learn the complete truth. Statements of co-conspirators are admissible on an agency theory. See, e.g., United States v. Russo, 302 F.3d 37, 45 (2d Cir.2002). We have explained that “[w]hen two persons engage jointly in a partnership for some criminal objective, the law deems them agents for one another. Each is deemed to have authorized the acts and declarations of the other undertaken to carry out their joint objective.” Id. (citing Anderson v. United States, 417 U.S. 211, 218 n. 6, 94 S.Ct. 2253, 41 L.Ed.2d 20 (1974) (“The rationale for both the hearsay-conspiracy exception and its limitations is the notion that conspirators are partners in crime. As such, the law deems them agents of one another.”)). It follows then that having formed their conspiracy to obstruct the ImClone investigation, Stewart and Bacanovic, each having full authority as an agent to speak for the other on matters within the purview of the conspiracy, answered questions on each other’s behalf during the questioning by the Government’s investigators and attorneys. The testimony by one, therefore, was as a matter of law the authorized statement of the other. For these reasons, we hold that when the object of a conspiracy is to obstruct justice, mislead law enforcement officers, or commit similar offenses by making false statements to investigating officers, truthful statements made to such officers designed to lend credence to the false statements and hence advance the conspiracy 4 are not rendered inadmissible by the Confrontation Clause. A contrary reading of the rule would result in obvious and unacceptable impediments to prosecuting cases like this one, in which the very object of the charged conspiracy is for the defendants to mislead investigators by responding falsely to the investigators’ questions in a structured setting, fully aware that their responses might be used in future judicial proceedings. For these reasons, there was no error here in admitting the testimonial statements of one Defendant against the other. 2. Admitting the challenged statements did not affect substantial rights Even assuming the first two elements of plain error, moreover, we could not conclude that Defendants’ substantial rights were adversely affected by admitting the challenged statements. An evaluation of the relevant factors — which include “the importance of the witness’ testimony in the prosecution’s case, whether the testimony was cumulative, the presence or absence of evidence corroborating or contradicting the testimony of the witness on material points, the extent of cross-examination otherwise permitted, and, of course, the overall strength of the prosecution’s case” — indicates that any Confrontation Clause error, although we do not perceive one, *294 was harmless. 5 Delaware v. Van Arsdall, 475 U.S. 678, 684, 106 S.Ct. 1431, 89 L.Ed.2d 674 (1986). The majority of the challenged statements concern the conduct of which both Defendants were acquitted, i.e., the charges that Stewart and Bacanovic lied about having reached an agreement prior to December 27th whereby Stewart’s ImClone shares would be sold if and when the per share price dropped to $60. Defendants do not demonstrate that they were prejudiced by the admission of evidence relating primarily to offenses on which the Government failed to carry its burden of proof. See United States v. Taylor, 92 F.3d 1313, 1332 (2d Cir.1996) (any error in admitting evidence primarily related to charge on which defendant was acquitted does not affect substantial rights and therefore is disregarded under Rule 52(b)). The counts of conviction were those relating to (i) Stewart’s having been informed about Waksal’s efforts to sell his ImClone holdings just prior to selling her own shares in the company; (ii) the message for -Stewart that Bacanovic left with Ann Armstrong on December 27th, and (iii) the Defendants’ communications thereafter about the investigation and the underlying events. Ample evidence, independent of the challenged statements, supports the jury’s verdict on these counts. With respect to the counts that were based on the Defendants’ communications on December 27th, the jury heard from Faneuil, whose credibility was exhaustively probed, that at Bacanovic’s direction he informed Stewart about Waksal’s efforts to sell his ImClone holdings and that Bacanovic later pressured him to cover up that fact when speaking to investigators, stressing to Faneuil that he and Stewart were “on the same page” and telling the “same story.” Faneuil also testified that he retained counsel because he had lied to the SEC. Faneuil told the jury that he used a colleague’s cell phone when calling Bacanovic to discuss the trades or the investigation in order to avoid Merrill Lynch taped lines; that testimony was corroborated by the phone records. In addition, phone records demonstrated that Bacanovic left a message for Stewart within minutes of learning from Faneuil of Waksal’s intent to sell. Stewart’s assistant Ann Armstrong testified that the message was “Peter Bacanovic thinks ImClone is going to start trading downward.” Phone records and the testimony of Waksal’s assistant, Emily Perret, demonstrated that after calling Bacanovic’s office in response to his message, Stewart called Waksal to learn what was going on with ImClone and insisted on speaking with him. The jury heard Armstrong describe Stewart’s temporary alteration of the phone log entry memorializing Baeanovie’s December 27th message. The jury listened to Stewart’s friend Mariana Pasternak recount Stewart’s remark that she was provided with the information about Waksal’s sale. As to the convictions on the counts that were based on Defendants’ misrepresentations about their communications with each other concerning the investigation, evidence indicated that Bacanovic sought a *295 meeting alone with Stewart that in fact took place. Faneuil stated that Bacanovic told him, shortly after that meeting, that he communicated with Stewart about the investigation and was assured that everyone was “on the same page.” On this record, there is also no basis to conclude that admitting the challenged statements, “seriously affect[ed] the fairness, integrity, or public reputation of judicial proceedings.” Johnson, 520 U.S. at 467, 117 S.Ct. 1544. In sum, we decline to reverse the judgments of conviction based on the argument that they were secured through testimony that violated Defendants’ Sixth Amendment rights. II. The interests of justice Defendants challenge the District Court’s denial of three separate motions for a new trial that were based on allegations of (i) false testimony by the Government’s ink expert, (ii) juror bias, and (iii) extraneous influences on the jury. A district court may vacate a judgment of conviction and grant a new trial “if the interest of justice so requires.” Fed.R.Crim.P. 33. We review the District Court’s decision to deny Defendants’ Rule 33 motions for abuse of discretion, upholding findings of fact that were made in the course of deciding the motions unless they are clearly erroneous. See, e.g., United States v. Wong, 78 F.3d 73, 78-79 (2d Cir.1996). A. Government expert’s perjury 1. Lawrence Stewart’s testimony Defendants each seek a new trial or, in the alternative, an evidentiary hearing, based on alleged prejudice arising from the introduction of false testimony given by the Government’s expert witness. During its case in chief, the Government called Lawrence F. Stewart, 6 a civilian employee of the United States Secret Service and its Laboratory Director and Chief Forensic Scientist, to give an expert opinion regarding the worksheet that Bacanovic described in his testimony before the SEC as having been prepared on December 20th when he and Stewart reviewed her portfolio. The worksheet is a one-page document that listed each stock in Stewart’s personal Merrill Lynch account and identified the market price as well as its unrealized gain or loss in position. Various handwritten notations, including circles, checks, and stock symbols appear on the page. In addition, to the right of the entry describing ImClone was the notation “@60.” Bacanovic testified on February 13, 2002, that the document demonstrated that Stewart sold her stock on December 27th pursuant to an earlier decision to liquidate if the per share price dropped to $60. The original worksheet was sent to the Secret Service Forensic Services Division (“FSD”) where the ink that produced the handwritten notations was analyzed. At trial, Lawrence described himself as the only “national expert for ink.” He testified that laboratory tests were performed — first in July and August 2002, and again in January 2004 — to determine whether the ink used in the “@60” notation was consistent with the ink in other notations on the page. Lawrence testified that the test results led him to conclude that the ink used to make the “@60” notation, which appeared next to the ImClone entry, differed from the ink used to make the other notations that he was able to test, all of which had been made with a type of Paper Mate pen. The Government *296 offered Lawrence’s testimony to demonstrate that the “@60” notation was not contemporaneous with the other notations on the page and therefore tended to prove that Baeanovic altered the document and that he and Stewart concocted the “@60” stop-loss order story after the fact when the flaws in the tax loss story came to light. The defense offered the testimony of Dr. Albert Lyter, a forensic chemist specializing in the analysis of inks and papers. There was little disagreement in the two experts’ conclusions. Both testified that it was likely that the marks other than the “@60” and a small dash at the end of the Apple Computer entry had been produced by a Paper Mate pen, that the ink used to make the “@60” mark was unusual and from an unidentifiable source, and that the timing of the various marks could not be determined reliably through testing. Lyter’s conclusion, however, departed from Lawrence’s in two respects. First, Lyter concluded that the “@60” notation and the Apple Computer dash had been made by the same pen, whereas Lawrence concluded that the “@60” ink differed from the ink used for other notations on the page but, because he had not tested the dash, he had made no conclusion regarding its source. Second, Lyter testified that use of a device known as a densitometer revealed batch variations, which indicated that at least two different ball point pens had been used to make the remaining notations on the page. The Government recalled Lawrence to answer questions about the use of densitometry. He testified that the densitometer was not sufficiently accurate to ascertain variations among different batches of the same ink recipe with reasonable reliability. Several months after the jury returned its verdict, the Government announced that an investigation had revealed that Lawrence had made false material statements in the testimony he gave in Stewart’s and Bacanovic’s trial. He was indicted on June 9, 2004 on two counts of perjury in violation of 18 U.S.C. § 1623 relating to his testimony that he had personally participated in the forensic tests about which he testified and that he was familiar with a book proposal drafted by his colleagues, and knew that it included a chapter on densitometry. Defendants contend that, in light of the perjury charges, reversal of their convictions should be “virtually automatic” because the Government knew or should have known at the time of trial that Lawrence’s testimony was false. They argue alternatively that the District Court erred in failing to hold an evidentiary hearing on the issue of the Government’s knowledge. We disagree. 2. Standard of review We have frequently acknowledged that, even where newly discovered evidence indicates perjury, motions for new trials “should be granted only with great caution and in the most extraordinary circumstances.” United States v. Sanchez, 969 F.2d 1409, 1414 (2d Cir.1992); accord United States v. Spencer, 4 F.3d 115, 118 (2d Cir.1993); United States v. DiPaolo, 835 F.2d 46, 49 (2d Cir.1987); United States v. Stofsky, 527 F.2d 237, 243 (2d Cir.1975); United States v. Costello, 255 F.2d 876, 879 (2d Cir.1958). So cautioned, the trial court’s discretion to decide whether newly discovered evidence warrants a new trial is broad because its vantage point as to the determinative factor— whether newly discovered evidence would have influenced the jury — has been informed by the trial over which it presided. See United States v. Gambino, 59 F.3d 353, 364 (2d Cir.1995) (recognizing that the trial court’s Rule 33 “ruling is deferred to *297 on appeal because, having presided over the trial, it is in a better position to decide what effect the newly discovered materials might have had on the jury”). 3. Analysis a. falsity “[WJhen the newly discovered evidence focuses on the perjury of a witness, a threshold inquiry is whether the evidence demonstrates that the witness in fact committed perjury.” United States v. White, 972 F.2d 16, 20 (2d Cir.1992). When the District Court denied Defendants’ requests for a new trial, it assumed that Lawrence perjured himself. Stewart, 323 F.Supp.2d at 615. At the time, Lawrence had been indicted on perjury charges arising from his testimony in this matter, but had not been convicted. Id. We will make the same assumption for purposes of this appeal, noting that the propriety of that approach is not altered by Lawrence’s subsequent acquittal, which does not establish that Lawrence’s trial testimony was true, but only that the Government did not prove beyond a reasonable doubt that his testimony was false. b. the Government’s awareness Perjury in and of itself is insufficient to justify relief under Rule 33. White, 972 F.2d at 22 (“the mere fact that [the witness] lied on the witness stand does not automatically entitle [defendant] to a new trial”). Rather, when a trial has been tainted by false testimony, this Court is “called upon to strike a fair balance between the need for both integrity and finality in criminal prosecutions” by determining whether false testimony was prejudicial in the sense that it affected the outcome of the trial. Stofsky, 527 F.2d at 239. To do so, we assess the materiality of the perjury to the verdict and are guided by two standards which are based on the extent of the government’s awareness of the false testimony prior to the conclusion of the trial. United States v. Wallach, 935 F.2d 445, 456 (2d Cir.1991). If the prosecution knew or should have known of the perjury prior to the conclusion of the trial, the conviction must be set aside where there is “any reasonable likelihood that the false testimony could have affected the judgment of the jury.” Id. (internal quotation marks omitted). The knowing introduction of false testimony will lead to “virtually automatic” reversal. Id. The standard applicable to the knowing introduction of .false testimony serves the dual purposes of discouraging prosecutorial misconduct and providing relief from an unfair conviction. See United States v. Agurs, 427 U.S. 97, 104, 96 S.Ct. 2392, 49 L.Ed.2d 342 (1976). On the other hand, if the prosecution was not aware of the perjury, a defendant can obtain a new trial only where the false testimony leads to “a firm belief that but for the perjured testimony, the defendant would most likely not have been convicted.” Wallach, 935 F.2d at 456 (internal quotation marks omitted). Defendants advance two theories for “virtually automatic” reversal of their convictions, arguing that the Government knew or should have known that Lawrence’s testimony was false. First, Defendants contend that, as part of the “prosecution team,” Lawrence’s own knowledge or that of his FSD colleagues should be imputed to the Government. They do not assert that evidence in the record demonstrates that the prosecutors actually knew that Lawrence was lying. Alternatively, Defendants contend that “red flags” should have alerted the prosecutors to Lawrence’s lies. i. imputed knowledge As the Government points out, this Court has not expressly recognized or re *298 jected the imputation principle in the context of a new trial motion based on evidence of perjured testimony. Nor do we have occasion to do so now. Without dismissing the possibility that there are circumstances in which it may be fair to impute the knowledge of certain persons to prosecutors as if it were their own, we find that the District Court did not err in concluding that those circumstances are not present here. Drawing on Supreme Court authority holding that Brady obligations extend to all persons “acting on the government’s behalf,” Defendants urge that the scope of the “prosecution team” should be similarly construed to impute government agents’ false testimony to the prosecutors themselves. Kyles v. Whitley, 514 U.S. 419, 437, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995); see also Wedra v. Thomas, 671 F.2d 713, 717 n. 1 (2d Cir.1982) (noting that “the knowledge of a police officer may be attributable to the prosecutor if the officer acted as an arm of the prosecution”). According to Defendants, because Lawrence was a “government official” working in conjunction with the prosecution, it is fair to attribute his knowledge of the perjured testimony, or that of other FSD employees, to the Government. But our determination of whether to deem an individual to be an “arm of the prosecution” for Brady purposes does not follow the broad, categorical approach urged by Defendants. Instead, the propriety of imputing knowledge to the prosecution is determined by examining the specific circumstances of the person alleged to be an “arm of the prosecutor.” United States v. Morell, 524 F.2d 550, 555 (2d Cir.1975). It does not turn on the status of the person with actual knowledge, such as a law enforcement officer, prosecutor or other government official. In other words, the relevant inquiry is what the person did, not who the person is. See id. (imputing law enforcement agent’s knowledge of confidential file to prosecutors where agent supervised the witness, participated actively in the investigation and frequently sat at counsel table throughout the trial); see also United States v. Sanchez, 813 F.Supp. 241, 247-48 (S.D.N.Y.1993) (imputing to prosecutor knowledge of perjury of local police officers who were deputized as federal agents and worked as part of investigative task force), aff'd on other grounds, 35 F.3d 673 (2d Cir.1994); Pina v. Henderson, 752 F.2d 47, 49 (2d Cir.1985) (refusing to apply the “descriptive term” of “arm of the prosecutor” to parole officer who did not work “in conjunction” with the police or the prosecutor). We do not agree with Defendants that the District Court dismissed the imputation argument as applicable only to law enforcement officials who were involved with the investigation. Rather, in finding that Lawrence acted as an ordinary expert witness and not as part of the “prosecution team,” the District Court properly analyzed what Lawrence actually did and did not do in connection with the investigation and subsequent judicial proceedings. Stewart, 323 F.Supp.2d at 616-18. The District Court’s factual findings, which are not clearly erroneous, demonstrate that Lawrence’s role was limited to matters concerning his area of expertise— ink. In that regard, Lawrence analyzed a single document, explained the forensic ink tests that had been performed, discussed potential testimony by the defense ink expert, assisted prosecutors to develop cross-examination questions addressing certain technical aspects of ink testing, and participated in a mock examination on ink issues to prepare for trial. His testimony at trial related only to his credentials, the tests that were performed and the conclusions he drew from them. None of this suggests *299 that Lawrence was in any way involved with the investigation or presentation of the case to the grand jury. He did not interview witnesses or gather facts, nor, with the exception of the “@60” worksheet, did he review documents or develop prosecutorial strategy. Lawrence acted only in the capacity of an expert witness, as the District Court found, and not as a “fully functioning member of the prosecution team,” as Defendants suggest. The fact that Lawrence was a government employee does not alter that conclusion, which applies with even more force to the other laboratory employees whose participation was similarly limited in scope and less extensive than Lawrence’s. These circumstances, therefore, do not justify attributing to the prosecutors, as if it was their own, knowledge that Lawrence or other FSD personnel possessed. ii. disregard of “red ñags” Alternatively, Defendants urge that even if the prosecutors did not in fact know that Lawrence lied, they should have been alerted to the possibility that he made misrepresentations as a result of certain matters that arose prior to and during the trial. We are not left with any inclination, much less a “ ‘definite and firm conviction,’ ” that the District Court made a mistake, United States v. Garcia, 413 F.3d 201, 222 (2d Cir.2005) (quoting United States v. United States Gypsum Co., 333 U.S. 364, 395, 68 S.Ct. 525, 92 L.Ed. 746 (1948)), when it found that the record does not indicate that prosecutors negligently ignored indications that Lawrence overstated his participation in initial ink tests. See Stewart, 323 F.Supp.2d at 618-19 (noting that Defendants do not contend that there were “red flags” that should have caused the Government to suspect that Lawrence would lie about being aware of his colleagues’ book proposal). The record fully supports the District Court’s finding that the Government did not fail to “properly utilize the available information” where (i) the absence of Lawrence’s name on the 2002 forensic report was not necessarily suspicious, nor was it inconsistent with his representation that he worked with another FSD employee, Susan Fortunato; (ii) Lawrence’s and Fortunato’s explanations of their initial failure to test the ink content of the Apple Computer dash did nothing to put prosecutors on notice that Lawrence .would later misrepresent the extent of his participation in the 2002 testing; and (iii) Fortunato’s description of an initial meeting with prosecutors does not establish that she informed them that she alone conducted the ink testing. Id. (quoting Wallach, 935 F.2d at 457). The District Court did not err in finding that the facts did not demonstrate that the Government knew or should have known that Lawrence’s testimony was false. There is no indication that the false testimony was introduced as a result of prosecutorial misconduct. c. materiality Ultimately, whether the prosecution was aware of the alleged perjury is of no moment, because we conclude that the testimony in question was not material under either standard. That is to say, whether we ask if the verdict “might have been different” or if it “probably would have been different” had the jurors known (i) the truth about Lawrence’s involvement in the ink testing and his awareness of his colleagues’ book proposal discussing densitometry and (ii) that Lawrence had lied under oath about those matters, the answer is no. Therefore, we conclude that the District Court did not err in finding that Lawrence’s false statements were not material either as to the factual elements *300 of the Government’s case or for impeachment value, had the misrepresentations been exposed at trial. See Stofsky, 527 F.2d at 246. Lawrence’s testimony did not influence the verdict on the counts of conviction. It pertained exclusively to Bacanovic’s “@60” worksheet and was used by the Government to support its position that the $60 stop-loss agreement was an after-the-fact fabrication. The jury acquitted Defendants of all of the counts and specifications relating to the existence of the agreement. Because the Government failed to persuade the jury to convict on the only counts to which Lawrence’s testimony related, that testimony cannot be considered capable of materially affecting the verdict on the counts to which it had no relevance. See White, 972 F.2d at 22 (determining that perjury was not material where defendant was acquitted of all charges that depended solely on perjurer’s testimony). Furthermore, the perjury was collateral to the substance of Lawrence’s own testimony — it did not concern the validity of the tests that were conducted or his conclusions, which were largely corroborated by the defense expert. Where, as here, “independent evidence supports a defendant’s conviction, the subsequent discovery that a witness’s testimony at trial was perjured will not warrant a new trial.” Wong, 78 F.3d at 82 (citing United States v. Reyes, 49 F.3d 63, 68 (2d Cir.1995)). The District Court found that “the jury convicted defendants of lies that had nothing to do with the $60 agreement.” Stewart, 323 F.Supp.2d at 619. We see no reason to disturb the District Court’s finding that ample evidence unrelated to Lawrence’s testimony or to the $60 stop-loss agreement supported the verdict on the counts of conviction. See, e.g., Wong, 78 F.3d at 78-79 (factual findings in connection with denial of motion for new trial upheld unless clearly erroneous). As to Martha Stewart, the verdict of conviction was based on proof beyond a reasonable doubt that Stewart lied to investigators (i) when she told them that she spoke to Bacanovic on December 27th and instructed him to sell her ImClone shares, (ii) she could not remember whether she was told about Waksal’s efforts to sell his shares, (iii) she sold the stock when she did so because she did not want to be bothered during her vacation, (iv) she discussed MSLO and K-Mart with Bacanovic on December 27th, and (v) she did not discuss the investigations with Bacanovic. See Stewart, 323 F.Supp.2d at 619. As to Bacanovic, the jury found that he lied when he told investigators that he spoke to Stewart on December 27th and when he described a message he left with Armstrong for Stewart that only gave the price of ImClone shares. See id. at 619-20. Like the District Court, we find that it is not possible, much less probable, that the jurors would have acquitted Stewart and Bacanovic of these charges and specifications had they known about Lawrence’s limited participation in the ink testing, his awareness of his colleagues’ book proposal or even his alleged attempt to cover up the real reason that the Apple Computer dash was not tested by FSD in July 2002. These simply cannot be matters that would “ ‘ereate[ ] a reasonable doubt that did not otherwise exist’” concerning the factual elements of the counts of conviction. United States v. Gambino, 59 F.3d 353, 365 (2d Cir.1995) (quoting Agurs, 427 U.S. at 112, 96 S.Ct. 2392). Nevertheless, Defendants advance a number of theories for concluding that revelation of the perjury would have resulted in a full acquittal, none of which is persuasive. Exposure of the false testimony would not have added impeachment value suffi *301 cient to change the outcome of the trial. Gambino, 59 F.3d at 364 (Rule 33 relief justified only if newly discovered evidence is of a type that could change the verdict).' Significantly, the District Court observed that the importance of the “@60” agreement and Lawrence’s testimony took on inflated importance as the “centerpiece” and “core theory” of the prosecution only after his false testimony came to light. Stewart, 323 F.Supp.2d at 620; cf. Wallach, 935 F.2d at 457-58 (determining that a new trial was necessary because lies of key witness who “tied all the pieces together,” even as to matters affecting only his credibility, could have caused the jury to reject his entire testimony and eliminate the foundation for conviction). Judge Cedarbaum “obviously was in the best position to appraise the possible effect” of the challenged testimony on the outcome of the trial because “[ajfter all, Judge [Cedarbaum] was at the trial. We were not.” United States v. Provenzano, 615 F.2d 37, 49 (2d Cir.1980). In a further effort to bridge the great gulf between Lawrence’s testimony and the counts of conviction, Defendants suggest that revelation of the perjury could have affected the verdict either by bolstering the jury’s confidence in Defendants’ position, exposing a lack of confidence by the Government in its own case or impeaching other Government witnesses by extension. These arguments are speculative and unpersuasive. Defendants offer nothing more than wishful conjecture that disclosure of Lawrence’s lies would have endowed the “@60” agreement with some amorphous plus factor capable of translating the jury’s implicit recognition of the existence of the “@60” agreement into a conclusion that it was the sole motivating factor for Stewart’s ImClone sale. In addition, without endorsing Defendants’ theory that Government fact witnesses could have been impugned by association with Lawrence, we note that newly discovered evidence of perjury that serves only to impeach credibility is generally insufficient to justify a new trial. See Reyes, 49 F.3d at 68. In any event, Defendants do not explain how Lawrence’s false testimony would have diminished the credibility of Glotzer, Farmer, or any of the other witnesses called by the Government. As a final point, we do not agree that the Government’s indictment of Lawrence for perjury constitutes a concession that his perjury was material. Defendants argue that having charged Lawrence with making “false material statements” at Defendants’ trial, in violation of 18 U.S.C. § 1623, the Government should be precluded from denying the materiality of the false statements to resist a Rule 33 motion. There is no authority for that proposition, and the consequence of Defendants’ reasoning would be that any prosecution of an individual for giving perjured testimony in a criminal proceeding would lead to a new trial regardless of whether the perjury had any effect on the jury’s verdict. That result is inconsistent with our rejection of a Rule 33 standard that would “require reversal when the perjury went only to an immaterial aspect of the proof.” Sanders v. Sullivan, 863 F.2d 218, 226 (2d Cir.1988) (reflecting upon Stofsky, 527 F.2d at 245, in which the Court discussed its discomfort with a test that “require[s] reversal in cases of perjury with respect to even minor matters”). The need for a new trial depends on the effect of the perjury on the verdict, see Stofsky, 527 F.2d at 243-47, whereas materiality for purposes of a section 1623 prosecution is “whether the false testimony was capable of influencing the fact finder in deciding the issues before [it],” United States v. Guariglia, 962 F.2d 160, 164 (2d Cir.1992) (internal quotation marks and citations omitted; alteration in original). *302 The standards for assessing materiality in the two contexts reflect the different purposes effectuated by the rule and the statute. The materiality inquiry in a Rule 33 motion focuses on the ultimate outcome of the trial because the objective of the rule is to relieve a defendant of the unfairness that results when false testimony contributes to a verdict of conviction. See Stofsky, 527 F.2d at 239 (affirming conviction where alleged perjury was not sufficiently significant to have “unfairly tainted” verdict). Section 1623, on the other hand, serves to deter and punish false testimony and may be based more broadly on any matter before the jury. See United States v. Reed, 773 F.2d 477, 483 (2d Cir.1985) (observing that section 1623 facilitates intent to deter perjury and enhance reliability of testimony); United States v. Gugliaro, 501 F.2d 68, 71-72 (2d Cir.1974) (“misstatement of fact need not be ‘dispositive’ of the inquiry in question to be material” in perjury prosecution (quoting United States v. Birrell, 470 F.2d 113, 115 n. 1 (2d Cir.1972))). In sum, we find that the interests of justice do not require a new trial because Lawrence’s false testimony did not result in unfairness to Defendants. The District Court did not err in finding that there is no reasonable possibility, much less a probability, that the jury’s decision to convict Defendants for lying about their various communications on and after December 27th would have been different if the jurors had known the facts that Lawrence concealed or had known that Lawrence lied under oath. Nor did the District Court abuse its discretion by denying Rule 33 relief without convening an evidentiary hearing to ascertain the extent of the Government’s awareness of the perjury. Where, as here, the additional evidence of perjury is not sufficiently material to undermine confidence in the verdict, there is no need to probe the extent of the Government’s awareness of the perjury because the argument fails under either standard. See White, 972 F.2d at 22 (holding that where “it is not necessary to resolve the issues that might be the focus of an evidentiary hearing, the district court did not abuse its discretion in refusing to conduct an evidentiary hearing”). The District Court’s denial of Defendants’ Rule 33 motions based on Lawrence’s testimony is affirmed in all respects. B. Jury bias and extraneous influence Defendants charge that the District Court erred by denying Rule 33 relief without holding an evidentiary hearing on alleged juror misconduct. Defendants claim that evidence indicating that juror Chappell Hartridge gave false voir dire responses justified further inquiry into his ability to be fair and impartial. Baeanovic makes the additional argument that a new trial, or at least an evidentiary hearing, is warranted because post-verdict public statements by Hartridge and other jurors demonstrate that matters outside of the record improperly influenced the verdict of conviction. 1. Standard of review Post-trial jury scrutiny is disfavored because of its potential to undermine “full and frank discussion in the jury room, jurors’ willingness to return an unpopular verdict, and the community’s trust in a system that relies on the decisions of laypeople.” Tanner v. United States, 483 U.S. 107, 120-21, 107 S.Ct. 2739, 97 L.Ed.2d 90 (1987). Accordingly, probing jurors for “potential instances of bias, misconduct or extraneous influences” after they have reached a verdict is justified “only when reasonable grounds for investigation exist,” in other words, where there is “clear, strong, substantial and incontro *303 vertible evidence that a specific, nonspeeulative impropriety has occurred which could have prejudiced the trial.” United States v. Moon, 718 F.2d 1210, 1234 (2d Cir.1983) (citing King v. United States, 576 F.2d 432, 438 (2d Cir.1978)). The inquiry should end whenever it becomes apparent to the trial judge that “reasonable grounds to suspect prejudicial jury impropriety do not exist.” Id. We review for abuse of discretion the District Court’s Rule 33 decision regarding the effect on the jury of “potentially prejudicial occurrences.” Id. at 1235. 2. Analysis a. false voir dire responses Reviewing the District Court’s denial of Rule 33 relief based on alleged juror bias, we are mindful of the principle that [o]ne touchstone of a fair trial is an impartial trier of fact — “a jury capable and willing to decide the case solely on the evidence before it.” Smith v. Phillips, 455 U.S. 209, 217, 102 S.Ct. 940, 71 L.Ed.2d 78 (1982). Voir dire examination serves to protect that right by exposing possible biases, both known and unknown, on the part of potential jurors. Demonstrated bias in the responses to questions on voir dire may result in a juror being excused for cause; hints of bias not sufficient to warrant challenge for cause may assist parties in exercising their peremptory challenges. The necessity of truthful answers by prospective jurors if this process is to serve its purpose is obvious. McDonough Power Equip., Inc. v. Greenwood, 464 U.S. 548, 554, 104 S.Ct. 845, 78 L.Ed.2d 663 (1984). These concerns are reflected in the McDonough standard for analyzing allegations that a juror’s false voir dire concealed bias that affected the fairness of the trial: a party alleging unfairness based on undisclosed juror bias must demonstrate first, that the juror’s voir dire response was false and second, that the correct response would have provided a valid basis for a challenge for cause. See McDonough, 464 U.S. at 556, 104 S.Ct. 845. The District Court observed that this Court has never found reason to overturn a verdict on the basis of juror nondisclosure under McDonough and only once, see United States v. Colombo, 869 F.2d 149 (2d Cir.1989), has remanded for an evidentiary hearing on the matter. See United . States v. Stewart, 317 F.Supp.2d 432, 437 (S.D.N.Y.2004). Our ruling here does not change those statistics. Prompted by the extensive publicity surrounding this case, the District Court undertook a two-step voir dire process. First, prospective jurors completed a questionnaire drafted by the parties. Then, following for-cause challenges based on the questionnaire responses, those persons remaining in the jury pool were questioned individually. The questionnaire probed prospective jurors’ prior involvement with the justice system by asking about court appearances and whether the individual or someone close to him or her had filed criminal charges, had been the victim of a crime, had been sued, accused of wrongdoing on a job, or questioned by law enforcement or accused of, charged with, or convicted of any crime. Hartridge’s questionnaire reflected that he had appeared in court in connection with a dispute with his landlord, but he gave negative responses to the other questions described above. He returned for individual questioning and was subsequently empaneled. Shortly after the jury’s verdict was announced, Hartridge and several other jurors discussed' their experience as jurors with the press. Ac *304 cording to Defendants, at that time “the defense received information calling into question the manner in which Hartridge got on the jury.” In their motions for a new trial, Defendants alleged that Hartridge’s failure to disclose the following five matters on his questionnaire concealed probable bias: (i) an arrest and arraignment for assault of a former girlfriend; (ii) civil suits against Hartridge and members of his family; (iii) his son’s conviction for attempted robbery; (iv) an accusation of embezzlement in his capacity as a Little League volunteer treasurer; and (v) termination for cause from employment with Citibank. As support, Defendants proffered affidavits, only one of which was based on personal knowledge, and court records. The District Court found that the majority of Defendants’ allegations did not satisfy the first prong of the McDonough test because they rested on “little more than hearsay, speculation, and in one instance, vague allegations made by a person who refused to identify himself.” Stewart, 317 F.Supp.2d at 438. The District Court also found that certain ambiguities in the voir dire questions developed by the parties made it unclear that Hartridge’s responses deliberately concealed the truth. Id. As to those alleged failures to disclose, this was not the type of showing that constitutes “reasonable grounds” for investigation. Moon, 718 F.2d at 1234. The District Court credited Defendants with raising serious questions concerning deliberate concealment of two matters that Hartridge did not disclose, namely, the conviction of Hartridge’s son for attempted robbery in 2000 and a civil judgment against Hartridge and his wife in 1997, both of which were supported by objective evidence. Stewart, 317 F.Supp.2d at 441-42. But the District Court found that none of the alleged omissions satisfied the second part of the McDonough test, which requires the party seeking a new trial to demonstrate that the correct answer at jury selection would have provided a valid basis to challenge the prospective juror for cause. McDonough, 464 U.S. at 556, 104 S.Ct. 845. To do so, the district court must “determine if it would have granted the hypothetical challenge.” United States v. Greer, 285 F.3d 158, 171 (2d Cir.2002). Its determination is reviewed for abuse of discretion and, in that regard, we have noted that “ ‘[tjhere are few aspects of a jury trial where we would be less inclined to disturb a trial judge’s exercise of discretion, absent clear abuse, than in ruling on challenges for cause in the empanelling of a jury.’ ” Id. at 172 (quoting United States v. Ploof, 464 F.2d 116, 118 n. 4 (2d Cir.1972)). Defendants contend that the juror’s multiple failures to disclose background information indicate a suspicious pattern sufficient to raise doubts about his impartiality which justifies further inquiry in a hearing. For this proposition, Defendants cite a decision in which the Ninth Circuit observed that certain false statements that “might be harmless in isolation” may present a “much more sinister picture” when viewed as a whole. Green v. White, 232 F.3d 671, 678 n. 10 (9th Cir.2000). The assertion that Hartridge engaged in a pattern of lies ignores the District Court’s finding, which is not clearly erroneous, that only two of the alleged intentional omissions were supported with the type of evidence that justifies further inquiry. In Green, the Ninth Circuit found error in the trial court’s finding of no bias where the juror not only lied twice about his background but gave misleading and false explanations when confronted with the lies and also made comments (ie., that he knew the defendant was guilty on first sight) that left little doubt as to his inability to render a fair verdict. Id. at 677-78. In this case, the *305 allegations form a pattern that might create “destructive uncertainties” about Hartridge’s indifference if they were proved, id. at 676 (quoting Dyer v. Calderon, 151 F.3d 970, 983 (9th Cir.1998) (in banc)), but the Defendants’ proffer is insufficient as to all but two of those allegations. 7 The significant factor, however, is that the District Court found that even if it were established that Hartridge’s responses were false as alleged, none of the correct answers would have supported an inference that he was biased or prejudiced against Stewart and Bacanovic or had prejudged the evidence. See Stewart, 317 F.Supp.2d at 443. Nor is Colombo to the contrary. In Colombo, a district court’s decision to refuse Rule 33 relief on the basis of alleged false voir dire responses was vacated and the case was remanded for an evidentiary hearing. Colombo, 869 F.2d at 150. The alleged false responses asserted by Stewart and Bacanovic differ distinctly from the ones at issue in Colombo. There, the defendant’s Rule 33 motion was supported by an alternate juror’s affidavit, which stated that one of the jurors (i) confided that she failed to disclose that her brother was a government attorney because she feared the relationship would make her ineligible to serve on the jury and (ii) asserted familiarity with one of the conspirators’ meeting places as a “hang out for gangsters.” Id. In Colombo, “it was not simply that the lies in question were deliberate, but that the deliberateness of the particular lies evidenced partiality.” Greer, 285 F.3d at 172-73 (discussing Colombo ). Thus, where the Colombo affiant specifically asserted that the juror admitted not only concealment, but concealment of facts from which specific bias could be inferred, the alleged lies, if established, would demonstrate both dishonesty and partiality and would satisfy both prongs of the McDonough test. Id. That is not the case here. Defendants note that all of the jury bias cases cited by the District Court involved post-trial hearings. See Greer, 285 F.3d at 166; United States v. Shaoul, 41 F.3d 811, 814-15 (2d Cir.1994); United States v. Langford, 990 F.2d 65, 67 (2d Cir.1993). In the present case, the District *306 Court did not abuse its discretion in refusing to conduct a hearing because it found that even if all of the allegations were established by competent evidence, none would provide valid grounds to challenge for cause Hartridge’s membership on the jury. While an evidentiary hearing “is not held to afford a convicted defendant the opportunity to ‘conduct a fishing expedition,’ ” Moon, 718 F.2d at 1234 (quoting United States v. Moten, 582 F.2d 654, 667 (2d Cir.1978)), it is possible, even plausible, that a district court’s decision to hold a hearing under these circumstances would not have licensed such an expedition. We therefore caution district courts that, if any significant doubt as to a juror’s impartiality remains in the wake of objective evidence of false voir dire responses, an evidentiary hearing generally should be held. Cfi United States v. Boney, 977 F.2d 624, 634 (D.C.Cir.1992) (recommending that an evidentiary hearing be held under most circumstances, including a juror’s lying about his felony status, but declining to hold that “any false statement or deliberate concealment by a juror necessitates an evidentiary hearing”). Such a hearing is often the most reliable way for discerning the true motivations behind a juror’s false replies. See Smith, 455 U.S. at 215, 102 S.Ct. 940 (“[T]he remedy for allegations of juror partiality is a hearing in which the defendant has the opportunity to prove actual bias.”). Here, however, although we might have ruled differently on the hearing request in the first instance, we cannot say that the District Court’s decision was an abuse of discretion. See Moon, 718 F.2d at 1235. b. consideration of extra-record evidence During the course of a televised interview shortly after the verdict was returned, jurors were asked what they thought about the testimony of Stewart’s friend, Mariana Pasternak. Their responses give rise to Bacanovic’s argument that the jury impermissibly considered matters outside the record in reaching the verdict as to him. During direct examination by the Government, Pasternak testified that while vacationing with Stewart at the end of December 2001, she “recalled] Martha saying that [Sam Waksal’s] stock is going down, or went down, and I sold mine.” When asked whether she recollected speaking with Stewart about brokers during the vacation, Pasternak replied, “I remember one brief statement, which was: ‘Isn’t it nice to have brokers who tell you those things.” ’ Pasternak was then asked if she knew who Stewart’s broker was and she replied “I know that Peter Bacanovic was Martha’s broker.” The District Court instructed the jury that the testimony could be considered only in connection with the charges against Stewart. On cross-examination, Pasternak stated that she could not remember with certainty whether the statements she attributed to Stewart were actually made by Stewart or whether they were Pasternak’s own thoughts. Bacanovic’s motion for a mistrial was denied, and prior to cross-examination of Pasternak, the District Court instructed the jury as follows: Testimony about what Martha Stewart told Ms. Pasternak is received in evidence only with respect to Martha Stewart. None of the statements of Martha Stewart to Ms. Pasternak that you heard yesterday afternoon are received in evidence against Peter Bacanovic, and it is not evidence against Peter Bacanovic. So that remember I told you at the beginning of the trial that guilt is personal, that you must separately consider each defendant and each charge. In considering the charges against Peter Bacanovic, you may not consider the *307 testimony about those statements of Martha Stewart to Ms. Pasternak in any respect. They have no bearing as to Peter Bacanovic. The jury received similar limiting instructions in the jury charge at the close of evidence. Bacanovic now contends that certain public remarks by Hartridge and other jurors reveal that the jury did consider Pasternak’s testimony in connection with the charges against him. In the interview, jurors were asked the following questions about the portion of Pasternak’s testimony quoted above: “What did you think when she said that?” and “But you found that pretty incriminating?” Hartridge replied, “Oh, very. Yeah. It took down two people with one shot because she mentioned Peter’s name. ‘It’s good to have a broker to tell you things.” ’ Other jurors seemed to agree. Bacanovic asserts that these remarks indicate that his conviction was improperly influenced by extra-record prejudicial material, and that the District Court erred by denying his motion for a new trial and his request for an evidentiary hearing in holding, in part, that Hartridge’s statement was inadmissible under Fed.R.Evid. 606(b). See United States v. Stewart, 317 F.Supp.2d 426, 431-32 (S.D.N.Y.2004). Fed.R.Evid. 606(b) precludes a juror from testifying about deliberations and jurors’ mental processes in the course thereof, but allows testimony regarding extraneous prejudicial information that came to the jury’s attention. Bacanovic argues that the juror’s statement regarding Pasternak’s testimony falls within the exception to Rule 606(b) because (i) it was outside the record, as the record pertained to him and (ii) was highly prejudicial because it was the only evidence corroborating Faneuil’s testimony that Bacanovic was involved with telling Stewart that Waksal was attempting to sell his stock. According to Bacanovic, a ruling against him on this point would have the effect of barring, under Rule 606(b), testimony that the jury improperly considered evidence extraneous to one defendant in a multidefendant trial, where Rule 606(b) would permit such testimony from a juror regarding the same extraneous evidence if the defendant was tried alone. We do not agree with that articulation of the issue. In fact, a ruling in Bacanovic’s favor would enable an end-run around the well-settled proposition that jurors are presumed to follow instructions, see, e.g. United States v. Dimming, 297 F.3d 52, 59 (2d Cir.2002), by permitting inquiry into a matter at the core of jury deliberations protected by Rule 606(b), see United States Football League v. Nat’l Football League, 842 F.2d 1335, 1371 n. 24 (2d Cir.1988) (rejecting, under Rule 606(b), claim that jurors were improperly influenced by information not in evidence despite district court’s limiting instruction). In so deciding, we agree with the Eighth Circuit’s decision to reject an argument, similar to Bacanovic’s, that Rule 606(b) did not prohibit the trial court from inquiring about jurors’ alleged discussion of the defendant’s failure to testify. See United States v. Rodriguez, 116 F.3d 1225 (8th Cir.1997). Like Bacanovic, the Rodriguez defendant argued that his failure to testify “was not evidence and should not have been considered, it should be considered an ‘outside influence’ about which the jurors should be allowed to testify.” Id. at 1226-27. The Eighth Circuit responded, That [defendant] did not testify is not a fact the jurors learned through outside contact, communication, or publicity. It did not enter the jury room through an external, prohibited route. It was part of the trial, and was part of the information each juror collected. It should not have been discussed by the jury, and *308 indeed was the subject of a jury instruction to that effect. But it was not “extraneous information,” and therefore does not fall within the exception outlined in Rule 606(b). Id. at 1227. We reach the same conclusion, for the same reasons. Here, as in Rodriguez, the alleged impermissible influence was not the result of information “the jurors learned through outside contact, communication, or publicity” or that “enter[ed] the jury room through an external, prohibited route.” Id. Rather, Pasternak’s testimony came “ ‘from the witness stand in a public courtroom where there is full judicial protection of the defendant’s right of confrontation, of cross-examination, and of counsel.’ ” Loliscio v. Goord, 263 F.3d 178, 185 (2d Cir. 2001) (quoting Turner v. Louisiana, 379 U.S. 466, 472-73, 85 S.Ct. 546, 13 L.Ed.2d 424 (1965)). As such, the testimony cannot be considered extraneous information. 8 Accordingly, in considering whether Bacanovic’s Sixth Amendment rights were implicated by an improperly influenced jury, the District Court did not abuse its discretion by refusing to order a new trial or an evidentiary hearing that was clearly proscribed by Rule 606(b). III. Jury instructions and evidentiary rulings Stewart and Bacanovic argue that they were prejudiced by the specter of an uncharged offense. They assert that the trial was infected by language and evidence concerning insider trading. The dispute over this issue arose early in the proceedings, when Stewart moved to strike portions of the Superseding Indictment, alleging that it invoked facts, theories and terminology that were relevant only to the crime of insider trading, which was not charged. The Government moved in limine to preclude the defense from arguing that jurors could draw an inference from the fact that Stewart was not charged with insider trading. The District Court granted the Government’s motion and established the following ground rules for trial: The Government seeks to prevent defendants from inviting the jury to speculate about why the indictment does not charge the crime of insider trading. Clearly, defendants may inform the jury that the indictment does not charge them with the crime of insider trading. But defendants may not invite the jury to speculate as to why that charge was not included in the indictment. Nor may they argue that the absence of an insider trading charge proves their innocence of such activity. If the Government presents arguments or evidence that tend to show that defendants were motivated not only by the fear that they would be accused of trading illegally, but also that such a fear was justified — that is, that Stewart’s trading was illegal — then it will open the door to defense evidence that the conduct was not illegal. United States v. Stewart, No. 03 CR. 717(MGC), 2004 WL 113506, *1-2 (S.D.N.Y. Jan.26, 2004). Accusations by *309 each side that the other side failed to comply with the ruling were first lodged during opening statements and repeated throughout the trial. In rulings made in connection with the issue, the District Court drew a distinction between (1) Stewart’s state of mind, or belief, regarding the possibility that her ImClone sale would result in insider trading charges, and (2) whether the ImClone sale transaction was actually illegal as a matter of fact or law. The former was relevant to the charged offenses; the latter was not. Defendants now challenge the District Court’s refusal to instruct the jury that Defendants were not charged with insider trading. They also argue that the District Court made erroneous evidentiary rulings, including preventing Stewart’s securities law expert from testifying, that unfairly restricted their right to present a complete defense. A. Failure to give requested jury instruction Defendants contend that the District Court erred by failing to instruct the jury, as proposed in Stewart’s Request Number 17, that Neither Martha Stewart nor Peter Bacanovic has been charged with Insider Trading in this case. Your deliberations are limited to the facts that have been proved in court during the course of this trial. Thus you may only consider that which was presented in court in relation to the charges in the Indictment. You may consider Insider Trading only insofar as it pertains to motive for the obstruction charge. The government contends that information about the Waksal sales was believed by Ms. Stewart to constitute inside information. Ms. Stewart vigorously denies that she so believed or that it was her intention to violate the law by trading on this type of information. You may not, and I caution you strongly against this, you may not conclude that the government should have charged Ms. Stewart with Insider Trading and convict her of anything else in place of a charge that was not filed by the government and charged by the grand jury because it appeals to your sense of fairness or justice or what have you. You may not convict Martha Stewart unless there is sufficient evidence independent of Insider Trading to support a conviction on the charge in question. It is not clear that Defendants preserved this argument because the request was not discussed at the charge conference, nor was there any explicit objection to its omission. See Fed.R.Crim.P. 30 (a party objecting to the failure to give a requested instruction “must inform the court of the specific objection and the grounds for the objection before the jury retires to deliberate”). We need not resolve the waiver issue, however, because the argument fails whether we review the District Court’s decision de novo, see, e.g., United States v. Gonzalez, 407 F.3d 118, 122 (2d Cir.2005), or for plain error, see, e.g., United States v. Crowley, 318 F.3d 401, 414 (2d Cir.2003). In charging the jury, the District Court identified each specific charge against each Defendant and instructed the jury as to the elements essential to convicting each Defendant on each count as well as the necessity that the elements be proved beyond a reasonable doubt to support a guilty verdict. In addition, the jury received a copy of the Redacted Superseding Indictment and a verdict sheet that identified each specification in each count against each Defendant. The jury was directed to weigh the evidence and reach a verdict based solely on a consideration of the evidence and the instructions, and the *310 District Court stressed that “it would violate your sworn duty to base a verdict upon any other view of the law than that which I give you.” “‘A conviction will not be overturned for refusal to give a requested charge ... unless that [requested] instruction is legally correct, represents a theory of defense with basis in the record that would lead to acquittal, and the theory is not effectively presented elsewhere in the charge.’ ” United States v. Holland, 381 F.3d 80, 84 (2d Cir.) (quoting United States v. Han, 230 F.3d 560, 565 (2d Cir. 2000) (alteration in original)), cert. denied, 543 U.S. 1075, 125 S.Ct. 921, 160 L.Ed.2d 814 (2005). We “look at the charge as a whole to see if it correctly stated the law,” United States v. Jones, 30 F.3d 276, 284 (2d Cir.1994), and “will not find reversible error unless a charge either failed to inform the jury adequately of the law or misled the jury as to the correct legal rule,” United States v. Alfisi, 308 F.3d 144, 148 (2d Cir.2002). Defendants do not cite any precedent for finding that a trial was unfair because the jury did not receive an instruction not to consider culpability for specified uncharged crimes, where the instructions were otherwise proper. The District Court’s instructions were sufficient to restrict the jury’s use of the evidence to the crimes charged. It is clear from the partial verdict of acquittal that the jury “carefully evaluated the evidence and rendered [a] discriminating verdict[ ]” and not one that was based on uncharged acts or bad character. United States v. Casamento, 887 F.2d 1141, 1153 (2d Cir. 1989); see also United States v. Diaz, 922 F.2d 998, 1007-08 (2d Cir.1990) (partial acquittal belies notion of spillover prejudice). Certainly, the theory that Defendants should not be convicted for uncharged conduct was effectively presented elsewhere in the charge, namely in the portion devoted to explaining the elements that must be found beyond a reasonable doubt in order to convict. See Holland, 381 F.3d at 84. The District Court gave clear instructions as to the charges against each Defendant and the type and quantum of evidence on which to base a guilty verdict. We presume that juries follow their instructions, and the record does not suggest that the jurors were either confused or prejudiced against Defendants in determining the counts of conviction. See, e.g., United States v. Geibel, 369 F.3d 682, 700 (2d Cir.), cert. denied sub nom. Allen v. United States, 543 U.S. 999, 125 S.Ct. 619, 160 L.Ed.2d 457 (2004) and Conner v. United States, —U.S.-, 125 S.Ct. 1827, 161 L.Ed.2d 732 (2005). On this record, therefore, we find that the District Court did not err by omitting Stewart’s Request Number 17, much less commit plain error. See United States v. Carr, 424 F.3d 213, 221 (2d Cir.2005) (“There being no error, it follows a fortiori that there was no plain error.”). B. Evidentiary rulings Defendants assert that several of the District Court’s evidentiary rulings interfered with their rights to present a meaningful defense by calling witnesses on their behalf and cross-examining prosecution witnesses. Most of the challenged rulings pertain to the insider trading issues. These rulings precluded Defendants from calling a securities law expert, and imposed limitations on direct examination of Stewart’s business manager, Heidi DeLuca, and on cross-examination of Faneuil. “The [D]ue [P]rocess [CJlause of the Fifth Amendment and the compulsory process clause of the Sixth Amendment guarantee each criminal defendant the right to present a defense.” United States *311 v. Almonte, 956 F.2d 27, 30 (2d Cir.1992) (per curiam). It is equally well-settled that the right is subject to the application of procedural and evidentiary rules. See, e.g., Wade v. Mantello, 333 F.3d 51, 58 (2d Cir.2003) (collecting cases). Restrictions on presenting evidence do not offend the Constitution if they serve “legitimate interests in the criminal trial process” and are not “arbitrary or disproportionate to the purposes they are designed to serve.” Rock v. Arkansas, 483 U.S. 44, 55-56, 107 S.Ct. 2704, 97 L.Ed.2d 37 (1987) (internal quotation marks omitted). Similarly, the right to confront and cross examine witnesses is tempered by a trial judge’s “wide latitude” to impose “reasonable limits” in order to avoid matters that are confusing or of marginal relevance. Howard v. Walker, 406 F.3d 114, 128-29 (2d Cir.2005) (quoting Van Arsdall, 475 U.S. at 679, 106 S.Ct. 1431). Clearly, “[t]he accused does not have an unfettered right to offer testimony that is incompetent, privileged, or otherwise inadmissible under standard rules of evidence.” Taylor v. Illinois, 484 U.S. 400, 410, 108 S.Ct. 646, 98 L.Ed.2d 798 (1988). We review a trial court’s evidentiary rulings, including the decision to admit or exclude expert testimony, for abuse of discretion. See, e.g., Nimely v. City of New York, 414 F.3d 381, 393 (2d Cir.2005); see also United States, v. Anglin, 169 F.3d 154, 162 (2d Cir.1999) (“rulings on relevance under Rule 401 and admissibility under Rule 403 [reviewed] for abuse of discretion”); United States v. Duncan, 42 F.3d 97, 101 (2d Cir.1994) (the trial court’s discretion to admit or exclude expert testimony will not be disturbed unless “manifestly erroneous”). As we recently reiterated, a trial court acts within its discretion when it imposes restrictions that are not arbitrary or disproportionate to purposes designed to accommodate legitimate interests in the criminal trial process. See Howard, 406 F.3d at 132 (citing Rock, 483 U.S. at 55-56,107 S.Ct. 2704). 1. Exclusion of testimony Defendants contend that they were unfairly restricted from putting on a defense to the purported uncharged accusations of insider trading when the District Court prevented a securities law expert from testifying about the legality of Stewart’s ImClone trade. “Generally, the use of expert testimony is not permitted if it will ‘usurp either the role of the trial judge in instructing the jury as to the applicable law or the role of the jury in applying that law to the facts before it.’ ” Duncan, 42 F.3d at 101 (quoting United States v. Bilzerian, 926 F.2d 1285, 1294 (2d Cir.1991)). The proffered testimony regarding the legal principles underlying insider trading and their application to Stewart’s trade had the potential to invade both prohibited areas. Clearly, an opinion that purports to explain the law to the jury trespasses on the trial judge’s exclusive territory. 9 See, e.g., United States v. Scop, 846 F.2d 135, 139 (2d Cir.) (Fed.R.Evid. 704 “was not intended to allow experts to offer opinions embodying legal conclusions”), rev’d in part on reh’g on other grounds, 856 F.2d 5 (2d Cir.1988). In addition, the expert’s opinion that Stewart’s conduct did not violate the securities laws would have increased the potential for confusion because the facts supporting an insider trading *312 charge were not relevant to the charges under consideration. 10 Ultimately, the legality of trading upon the knowledge that Waksal was selling his ImClone holdings is irrelevant to the issue of whether Stewart later lied to investigators about having received that information. See Fed.R.Evid. 401 (relevant evidence tends to make the existence of a fact more or less probable). Nor would an opinion to that effect have assisted the jurors to understand the evidence or determine a factual issue. See Fed R. Evid. 702 (a witness qualified by the court as an expert may testify to an opinion that “will assist the trier of fact to understand the evidence or to determine a fact in issue”). Accordingly, preclusion of the expert witness was a proper exercise of the District Court’s discretion and did not violate Defendants’ right to present a defense. Similarly appropriate was the District Court’s exclusion of a portion of Heidi DeLuca’s testimony regarding a conversation she had with Faneuil concerning the impact of the ImClone trade on Stewart’s tax loss selling plan. Baeanovic asserts that he was unfairly prevented from eliciting from DeLuca that she recalled speaking with Faneuil about the ImClone gain in February 2002. Fanueil testified that the conversation took place in early January, giving rise to the Government’s argument that Defendants had to abandon the tax loss selling story and invent the $60 stop-loss story. The substance of DeLuca’s testimony about her conversation with Faneuil was properly excluded as hearsay. That ruling certainly resulted in no harm to Baeanovic because testimony from De-Luca about the date of the conversation was already in the record and because her testimony was relevant only to the counts concerning the legitimacy of Defendants’ pre-existing decision to sell ImClone at $60, of which they were acquitted. The District Court properly applied evidentiary rules that serve “legitimate interests in the criminal trial process,” and the resulting restrictions on the presentation of evidence were neither arbitrary nor disproportionate to those purposes. Rock, 483 U.S. at 55, 107 S.Ct. 2704 (internal quotation marks omitted). Defendants were not deprived of their constitutional rights to present a defense. 11 2. Limitations on cross-examination Defendants also claim that the District Court unreasonably restricted their rights to challenge prosecution witnesses. Stewart asserts that the District Court prevented her from cross-examining Faneuil fully about whether he thought that his conduct on December 27th violated the law, a matter she contends was critical to her defense against the uncharged crime of insider trading. Specifically, she asserts that she should have *313 been permitted to probe Faneuil regarding his understanding of a theory of liability based on misappropriation of client information. Stewart alleges that one of the ways that the Government “opened the door” to insider trading was to elicit testimony from Faneuil that he believed he did something illegal on December 27th, ie., he misappropriated client information; he “told one client about what another client was doing in his account and then lied about it to cover up.” In a line of questioning that focused on Faneuil’s belief that there was something wrong with Stewart’s ImClone transaction, Stewart’s counsel elicited testimony that Faneuil “didn’t feel comfortable with the situation,” that he “suspected” Stewart was precluded from trading on the information he gave her, and that twice on December 27th he involved himself in an illegal trade. We agree with the District Court that “the legal basis for [Faneuil’s] state of mind, whatever it was, whether it was because of the misappropriation theory or some other legal theory, is entirely beside the point.” The District Court’s decision to sustain an objection to questioning Faneuil on his understanding of the misappropriation theory was a proper exercise of its discretion to exclude irrelevant evidence and is not inconsistent with the right to confrontation. See, e.g., Howard, 406 F.3d at 129. Nor did the District Court abuse its discretion by deciding, pursuant to Fed.R.Evid. 403, that cross-examination of Faneuil on the subject of whether his marijuana use during a 2003 trip to Jamaica violated his cooperation agreement. Bacanovic sought to raise the matter to illustrate Faneuil’s incentive to lie to avoid termination of his cooperation agreement. A “mini-trial” on the legality of marijuana use in Jamaica and whether or not Faneuil could be prosecuted for his conduct or had breached his cooperation agreement would have been a confusing distraction in an already lengthy and complex trial. “In determining whether a trial judge has abused his discretion in the curtailment of cross-examination of government witnesses, the test is whether the jury was already in possession of sufficient information to make a discriminating appraisal of the particular ■ witness’s possible motives for testifying falsely in favor of the government.” United States v. Singh, 628 F.2d 758, 763 (2d Cir.1980). Considering the plethora of impeachment material against Faneuil — including his drug use, which featured prominently in the defense summations — the District Court did not abuse its discretion by determining that the risk of confusing and misleading the jury with collateral matters outweighed the probative value of clearly cumulative impeachment evidence. The District Court properly exercised its discretion to exclude the foregoing matters from evidence for the purpose of keeping the record reasonably free of evidence that was cumulative, confusing, misleading or irrelevant to the issues the jury was to decide. The restrictions were reasonable and did not violate Defendants’ confrontation rights. In sum, Defendants are not entitled to have their convictions set aside based on the District Court’s failure to give the requested “insider trading” charge, its exclusion of expert testimony or the limitations it imposed on examination and cross examination of witnesses. IV. Bacanovic’s other arguments Bacanovic made several additional arguments that pertain to his case only. 12 Like *314 the challenges discussed above, they do not persuade us that Bacanovic is entitled to relief from the jury’s verdict. A. Joinder and severance Bacanovic contends that the District Court erred in denying his request for a separate trial as well as his alternative request that Count Nine be severed from the other counts in the Superseding Indictment. Reviewing de novo the District Court’s decision under Rule 8 of the Federal Rules of Criminal Procedure, we find no error in its determination that the Defendants and the offense charged .in Count Nine were properly joined. See United States v. Feyrer, 338 F.3d 110, 113 (2d Cir.2003). Rule 8(b) permits joinder of defendants who “are alleged to have participated in the same act or transaction, or in the same series of acts or transactions, constituting an offense or offenses,” Fed.R.Crim.P. 8(b), which we have interpreted to mean that joinder is proper when the charged offense is “unified by some substantial identity of facts or participants, or arise out of a common plan or scheme.” United States v. Attanasio, 870 F.2d 809, 815 (2d Cir.1989) (internal quotation marks omitted). Without joinder of Stewart and Bacanovic, this matter would have resulted in two essentially duplicate trials because of the common factual elements of the charges and the evidence offered to establish them. See United States v. Turoff, 853 F.2d 1037, 1044 (2d Cir.1988). As the District Court observed, “whether Mr. Bacanovic is tried with or without Ms. Stewart, the charges against him inextricably associate him with Ms. Stewart [and][s]eparate trials cannot change that.” Similarly, joinder of the securities fraud charges in Count Nine against Stewart was proper under Rule 8(a) because the charges were “based on the same act or transaction” as the other charges the jury was to consider. Although it was not alleged that Bacanovic participated in Stewart’s June 2002 public statements, on which Count Nine was premised, he figured prominently in the events on which those statements were based. Stewart’s culpability under Count Nine depended on proving the same facts that supported other charges against both Defendants, ie., what transpired on December 27, 2001 and the falsity of Defendants’ representations to investigators about those matters. Under the circumstances, a joint trial on all of the pending charges served the purposes of trial efficiency and economy of judicial resources that are reflected in Rule 8 and the preference in the federal courts for a joint trial of defendants who are jointly indicted. See, e.g., Feyrer, 333 F.3d at 114; United States v. Amato, 15 F.3d 230, 237 (2d Cir.1994). Even where joinder is proper under Rule 8, Fed.R.Crim.P. 14 permits the district courts to grant relief from potential prejudice when there is a “ ‘serious risk that a joint trial would compromise a specific trial right of one of the defendants, or prevent the jury from making a reliable judgment about guilt or innocence.’ ” Feyrer, 333 F.3d at 114 (quoting Zafiro v. United States, 506 U.S. 534, 539, 113 S.Ct. 933, 122 L.Ed.2d 317 (1993)). The District Court’s determination that the circumstances here did not present such a risk is entitled to considerable deference that is “virtually unreviewable” and may be overcome only by demonstrating substantial prejudice “so severe that his conviction constituted a miscarriage of justice and *315 that the denial of his motion constituted an abuse of discretion.” United, States v. Yousef, 327 F.3d 56, 150 (2d Cir.2003) (internal quotation marks and citations omitted). Bacanovic does not persuade us that his conviction was the consequence of such unfairness by virtue of the jury’s alleged improper consideration of evidence admitted against Stewart only. As discussed above, the District Court gave clear and appropriately timed instructions to restrict the jury’s consideration of Mariana Pasternak’s testimony to the charges against Stewart and to clarify that evidence introduced to support the Count Nine charges was not relevant to the charges against Bacanovic. These instructions, coupled with the instruction that required jurors to consider separately each individual Defendant and each individual charge, are sufficient as an alternative to severance to cure the risk of unfair prejudice. See Zafiro, 506 U.S. at 539-40, 113 S.Ct. 933; United States v. Hernandez, 85 F.3d 1023, 1029-30 (2d Cir.1996) (rejecting claim of confusion and spillover prejudice where trial court instructed jurors “to consider the evidence against each defendant individually for each count”). 13 Furthermore, the fact that the jury found Bacanovic guilty of less than all of the counts and specifications charged against him strengthens the conclusion that his conviction was not the product of a “miscarriage of justice.” See United States v. Morales, 185 F.3d 74, 83 (2d Cir.1999). The District Court did not err in concluding that joinder of the Defendants and inclusion of Count Nine in the indictment were proper under Rule 8, nor did it abuse its discretion by refusing to grant Bacanovic’s Rule 14 motion for a separate trial or severance of Count Nine. B. Proof of perjury Bacanovic also claims that his perjury conviction under Count Six cannot stand because it rested on the uncorroborated testimony of one witness. See United States v. Weiner, 479 F.2d 923, 926 (2d Cir.1973). “In prosecutions for perjury, ... it has long been the rule that a conviction may not be obtained solely on the uncorroborated oath of one witness.” Id. This is the so-called “two witness rule.” “Technically, the ‘two witness’ rule is a misnomer because the rule requires either the testimony of a second witness or other evidence of independent probative value.” United States v. Maultasch, 596 F.2d 19, 25 n. 9 (2d Cir.1979); see also Weiner, 479 F.2d at 926 (“The rule is satisfied by the direct testimony of a second witness or by other evidence of independent probative value, circumstantial or direct, which is ‘of a quality to assure that a guilty verdict is solidly founded.’ ” (quoting United States v. Collins, 272 F.2d 650, 652 (2d. Cir. 1959))). “[T]he independent evidence must, by itself, be inconsistent with the innocence of the defendant.” Id. (internal quotation marks and alteration omitted). However, “the corroborative evidence [need not], in itself, be sufficient, if believed, to support a conviction.” Id. at 927. “In other words the two-witness rule has not been construed to require the Government, in effect, to prove its case twice over.” Id. Bacanovic was convicted of one count of perjury specifying that he falsely *316 testified to the SEC that he recalled leaving a message with Stewart’s assistant Ann Armstrong on the morning of December 27, 2001, in which he gave the current ImClone quote and asked that Stewart call him back, but that he did not recall telling Armstrong that ImClone was dropping. To demonstrate that Bacanovic testified falsely, the Government offered two pieces of evidence, (i) Armstrong’s testimony that she recalled Bacanovic telling her that he wanted to talk to Stewart about ImClone and to tell her that he thought it would start trading downward and (ii) an entry in the computerized message log Armstrong maintained that read “Peter Bacanovic thinks ImClone is going to start trading downward.” Armstrong specifically denied that Bacanovic gave the current price at which ImClone was trading. In the course of deliberations, the jurors sent a note to Judge Cedarbaum inquiring whether the “testimony of one witness and the document produced in the normal course of business by that same witness at a different time and place [could] be considered as two separate pieces of evidence.” The note was followed by a second one, which read, “We are referring to Annie Armstrong’s testimony and the phone logs she kept in the normal course of business at MSLO. Can the phone logs be used to corroborate her testimony?” After hearing argument, the District Court instructed the jury that the phone logs could be used to corroborate the testimony. Reviewing the pertinent part of the record, we find that the District Court did not err by doing so. See, e.g., Carr, 424 F.3d at 218 (jury instructions reviewed de novo). Bacanovic argues that the phone log is not sufficiently independent of Armstrong’s testimony to satisfy the two-witness rule because the two pieces of evidence share a common source. In that regard, the case presents a close question, and one that has not been addressed in our prior decisions: Is a document that was received in evidence as a business record under Fed.R.Evid. 803(6) sufficiently independent to corroborate' the testimony of the person who created the document in order to ensure that a perjury conviction does not result from the word of one person against another? See Weiler v. United States, 323 U.S. 606, 609, 65 S.Ct. 548, 89 L.Ed. 495 (1945) (“Since equally honest witnesses may well have differing recollections of the same event, we cannot reject as wholly unreasonable the notion that a conviction for perjury ought not to rest entirely upon ‘an oath against an oath.’ ”). The Supreme Court has explained that “[t]wo elements must enter into a determination that corroborative evidence is sufficient: (1) that the evidence, if true, substantiates the testimony of a single witness who has sworn to the falsity of the alleged perjurious statement; (2) that the corroborative evidence is trustworthy.” Id. at 610, 65 S.Ct. 548. We have interpreted the standard to mean that corroborating evidence is sufficient if it tends to “substantiate that part of the testimony of the principal prosecution witness which is material in showing that the statement made by the accused under oath was false.” Weiner, 479 F.2d at 927-28. The phone log was introduced during Armstrong’s direct testimony without objection as a business record under Fed. R.Evid. 803(6). As such, its reliability and trustworthiness derive from the circumstances under which it was created, rather than the author’s recollection. We have observed that Business records are made reliable by “systematic checking, by regularity and continuity which produce habits of precision, by actual experience of business in relying upon them, or by duty to make *317 an accurate record as part of a continuing job or occupation.” Fed.R.Evid. 803(6) Advisory Comm. Note. For this reason, business records may be admitted notwithstanding the unavailability of the record’s author, so long as a “custodian or other qualified witness” testifies that the document was “kept in the course of a regularly conducted business activity and also that it was the regular practice of that business activity to make the [record].” United States v. Williams, 205 F.3d 23, 34 (2d Cir.2000) (quoting United States v. Scarpa, 897 F.2d 63, 70 (2d Cir.1990)); see also Phoenix Assocs. III v. Stone, 60 F.3d 95, 101 (2d Cir.1995) (stating that “[t]he custodian need not have personal knowledge of the actual creation of the document” to lay a proper foundation for the receipt). Adoption of Rule 803(6) followed reform efforts to “relax[ ] the requirement of producing as witnesses, or accounting for the nonproduction of, all participants” in the gathering and recording of ordinary business records. Fed.R.Evid. 803(6) Advisory Comm. Note. Parker v. Reda, 327 F.3d 211, 214-15 (2d Cir.2003) (alterations in original). Although the Government laid the foundation for admitting the phone log as a business record through Armstrong’s testimony, its admissibility did not depend on her status as the author of the entries. See id. The same qualities that made the phone log admissible under Rule 803(6) made the document independently probative of the content of Bacanovic’s message. The assurance of accuracy and contemporaneousness that characterize a business record distinguishes that category of document from personal notations, which have been found insufficient to corroborate the author’s testimony for purposes of the two-witness rule. See Weiner, 479 F.2d at 929 (notation in personal diary of witness who testified that he spoke by phone with the accused insufficient to establish that the accused falsely denied the call). It is clear from our prior decisions that a document used to refresh a witness’s recollection will not constitute corroborating evidence to support a perjury conviction. Weiner, 479 F.2d at 929 (rule not satisfied by witness’s notation of phone call used to refresh recollection); United States v. Freedman, 445 F.2d 1220, 1226 (2d Cir.1971) (rule not satisfied where document offered as corroboration merely reinforced witness’s recollection). Because the message log was not used to refresh Armstrong’s recollection or as a basis for her testimony, each of the two pieces of evidence retained its independent probative value. The District Court did not err in determining that the phone log entry noting the content of Bacanovic’s December 27, 2001 message, which was recorded at that time in the normal course of MSLO business, was sufficiently independent of Armstrong’s recollection of the message, which she recounted in testimony more than two years later, to satisfy the requirements of the two-witness rule. See Weiner, 479 F.2d at 928 (“[T]he two-witness rule is satisfied by corroborative evidence of sufficient content and quality to persuade the trier that what the principal prosecution witness testified to about the falsity of the accused’s statement under oath was correct.”). We are assured that Bacanovic’s conviction under Count Six is solidly founded and not merely the product of two differing recollections of the same event. See Weiler, 323 U.S. at 609, 65 SUt. 548. 14 *318 C. Sufficiency of evidence and jury instruction on Count Two The jury found Bacanovic guilty of violating 18 U.S.C. § 1001, as charged in Count Two, by falsely stating, as alleged in Specification Two of that Count, that “he had a conversation with Martha Stewart on December 27, 2001 in which he told Stewart that ImClone’s stock price had dropped and Stewart told him to sell her ImClone stock.” Bacanovic now advances two theories for reversal of his conviction under Count Two. First, Bacanovic claims that the evidence supporting the conviction under Count Two was insufficient to establish materiality or intent, essential elements of section 1001 offenses. See 18 U.S.C. § 1001(a); United States v. Whab, 355 F.3d 155, 157-58, 162 (2d Cir.2004). In addressing a challenge to the sufficiency of evidence supporting a conviction, we review the evidence in the light most favorable to the prosecution, drawing permissible inferences in its favor, and will not set aside a conviction that rests on evidence that would convince a reasonable juror that the crime charged was proved beyond a reasonable doubt. See, e.g., United States v. Zhou, 428 F.3d 361, 369-70 (2d Cir.2005). A false statement is material if it tends to or is capable of influencing the decision-making body to which it was addressed. See United States v. Gaudin, 515 U.S. 506, 509, 115 S.Ct. 2310, 132 L.Ed.2d 444 (1995); United States v. Klausner, 80 F.3d 55, 59 (2d Cir.1996). Bacanovic’s statement was capable of distracting government investigators’ attention away from Faneuil if they believed that the only parties to the December 27th call were Stewart and Bacanovic, and in that regard it may be considered material. Similarly without merit is Bacanovic’s argument that the Government failed to prove that his false statement was intentional because (i) his use of pronouns was misunderstood, (ii) Faneuil informed them correctly that he was on the call with Stewart or (iii) Bacanovic later testified to that fact. We agree with the decisions from other circuits that have concluded that there is no safe harbor for recantation or correction of a prior false statement that violates section 1001. See United States v. Sebaggala, 256 F.3d 59, 64 (1st Cir.2001); United States v. Meuli, 8 F.8d 1481, 1486-87 (10th Cir.1993); United States v. Fern, 696 F.2d 1269, 1275 (11th Cir.1983). Likewise, sufficient evidence of materiality and intentionality also existed for the jury to convict Bacanovic on the basis of concealment. The “essential issue” is whether Defendant knowingly and willingly falsified, concealed, or covered up information relevant to the investigation. United States v. Stephenson, 895 F.2d 867, 874 (2d Cir.1990). Defendant’s legal duty to be truthful under section 1001 included a duty to disclose the information he had regarding the circumstances of Stewart’s December 27th trade, even though he voluntarily agreed to speak with investigators. See id. Bacanovic’s contention that there was insufficient evidence of the investigators’ questions is without merit. Trial testimony indicated that the SEC had specifically inquired about Bacanovic’s knowledge of Stewart’s trades. As a result, it was plausible for the jury to conclude that the SEC’s questioning had triggered Bacanovic’s duty to disclose and that *319 ample evidence existed that his concealment was material to the investigation. Bacanovic also argues that the indictment was duplicitous and that the jury instruction was erroneous as to Count Two. Under section 1001, in the course of a government proceeding, it is a crime (1) to falsify, conceal or cover up a material fact, (2) to make a materially false statement, or (3) to make or use a document containing a materially false statement. See 18 U.S.C. § 1001(a). The several different types of fraudulent conduct proscribed by section 1001 are not separate offenses, as Bacanovic suggests; rather they describe different means by which the statute is violated. See United States v. Crisci, 273 F.3d 235, 238-39 (2d Cir.2001) (no duplicity issue where indictment charges several means of committing a single crime); United States v. Murray, 618 F.2d 892, 896 (2d Cir.1980) (several offenses in one count is duplicitous, several means of committing an offense in a single count is not). The District Court did not err by failing to instruct the jurors that they must agree unanimously as to which theory of the offense — false statement or concealment— supported the verdict. See Schad v. Arizona, 501 U.S. 624, 631, 111 S.Ct. 2491, 115 L.Ed.2d 555 (1991) (where a single count alleges that the defendant committed a charged offense by one or more specified means, “[w]e have never suggested that ... the jurors should be required to agree upon a single means of commission, any more than the indictments were required to specify one alone”) (plurality opinion). United States v. Diogo, 320 F.2d 898 (2d Cir.1963), does not require a different conclusion. Despite the presence of language that might appear favorable to Bacanovic on this point, when this Court said in dictum that false representation and concealment are “two distinct offenses,” id at 902, we were not addressing the issue of duplicitous pleading. Rather, this Court was discussing the differences between the proof necessary to establish each of the theories. See id. at 901-02. We noted that a conviction for false representation requires evidence of actual falsity, while concealment is proved by showing “wilful nondisclosure by means of a ‘trick, scheme or device.’ ” Id. at 902 (quoting 18 U.S.C. § 1001). In that case, defendants were charged with lying about their marital status to evade the immigration laws. Id. at 900. The Court agreed that, although the marriages may not have been entered in good faith, defendants’ representation that they were married was literally true. Faced with a failure to prove any false statement, the Government urged the Court to affirm the convictions “on a theory of concealment.” Id. at 909. But the Court rejected the argument because the jury had not been instructed properly on the proof necessary to convict on the grounds that defendants’ statements concealed material facts through the use of a “trick, scheme or device.” Id. Read in context, Diogo recognizes distinctions in the factual elements of a false statement theory and a concealment theory, but the case does not suggest that two separate counts — one charging false conduct and the other charging concealment — could be based on the same conduct. See United States v. UCO Oil Co., 546 F.2d 833, 835 n. 2 (9th Cir.1976) (discussing Diogo dictum and observing that it “goes no further than to indicate that the different means of violating section 1001 involve different elements of proof’). CONCLUSION We have considered Defendants’ remaining arguments and find them to be without merit. For the reasons set forth *320 above, we affirm the judgments of conviction of Stewart and Bacanovie, but remand the case to the District Court to consider whether to modify the sentence imposed upon Bacanovie in accordance with Booker and Crosby. Footnotes [1] The District Court attached copies of the Superseding Indictment and the verdict forms as appendices to its opinion. See United States v. Stewart, 323 F.Supp.2d 606, 624-36 (S.D.N.Y.2004). [2] The Confrontation Clause' of the Sixth Amendment provides, "In all criminal prosecutions, the accused shall enjoy the right ... to be confronted with the witnesses against him ....” U.S. CONST, amend. VI. [3] After argument in the pending case, Stewart called our attention to United States v. Logan, 419 F.3d 172 (2d Cir.2005), decided on August 15, 2005. Logan considered the admissibility of false alibi statements made by two co-conspirators to law enforcement officers. The statements were not offered for their truth but rather to corroborate a statement and testimony that the two co-conspirators were planning to put forth a false alibi. See id. at 178. After rejecting a Confrontation Clause objection on the ground that the challenged statements had not been offered for their truth, see id., the Court stated in dicta that it was rejecting the Government's alternative contention that the statements were not testimonial within the meaning of Crawford because they were statements of co-conspirators in furtherance of a conspiracy. See id. at 178-79. The conspiracy offense charged in Logan was conspiracy to commit arson. See id. at 174. Whatever the force of the Logan dicta with respect to a conspiracy of that sort, it has no application to a case like ours, where the offense is a conspiracy to obstruct justice by making the very presentation of false statements with which the challenged truthful statements were interwoven. [4] Bacanovic contends that Defendants' contradictory statements to authorities (e.g., Stewart said Bacanovic discussed only the Merrill investigation with her, but Bacanovic stated he discussed the SEC investigation with her) are not admissible because they impeded the conspiracy rather than furthered it. However, to be admissible under the co-conspirator exception it is enough that the statements were made with the intent to further the conspiracy's purpose. Rule 802(d) does not require actual furtherance. See United States v. Reyes, 798 F.2d 380, 384 (10th Cir.1986); see also United States v. Minicone, 960 F.2d 1099, 1110 (2d Cir.1992). [5] Because Crawford was decided after her trial, Stewart contends that we should apply a modified plain-error analysis, and that the Government bears the burden of proving that plain error did not affect substantial rights. See United States v. Viola, 35 F.3d 37, 41-42 (2d Cir.1994). However, the Supreme Court’s decision in Johnson has called into doubt the continuing viability of the modified plain-error approach. See United States v. Thomas, 274 F.3d 655, 668 n. 15 (2d Cir. 2001) (in banc). Regardless of which side bears the burden, we reach the conclusion that substantial rights have not been harmed. [6] Like the District Court, we will refer to Lawrence F. Stewart as "Lawrence” in this opinion in order to distinguish his name from Martha Stewart, who is no relation. See Stewart, 323 F.Supp.2d at 613. *305 son’s arrest" for attempted robbery in the second degree. Id. at 442. The Court held that [7] The first "questionable omission" cited by the District Court concerned a 1997 judgment for $11,393 against Hartridge and his wife. The Court found that, [a]ssuming Hartridge had notice of this suit, the size of the judgment and its comparatively recent date suggest that he may have deliberately omitted it when he stated that he had never been sued. But defendants offer no explanation as to how the fact that a court has entered a judgment against a prospective juror supports an inference that the individual would be biased against defendants in a completely unrelated case. This information would not have supported a for-cause challenge, and accordingly cannot support defendants’ Rule 33 motion. Stewart, 317 F.Supp.2d at 441-42. The second omission noted by the District Court concerned whether Hartridge “deliberately omitted information concerning his defendants have still failed to demonstrate how that information would have supported a for-cause challenge of Hartridge. It is difficult to see how information concerning Hartridge’s son's conviction for an attempted robbery — a crime unrelated to the crimes charged in this case — could justify an inference that Hartridge would be biased against these defendants. If anything, a prospective juror with a family member who had been convicted of a crime would more likely be considered biased in favor of criminal defendants. At any rate, defendants offer no authority for the proposition that this conviction would support a for-cause challenge. Defendants have failed to satisfy the second prong of the McDonough test with respect to this allegation. Id. [8] By contrast, each of the cases cited by Bacanovic involved a jury’s consideration of information that was not admitted into evidence at all, as opposed to information properly admitted into evidence but used for an improper purpose. See United States v. Hall, 85 F.3d 367, 369-70 (8th Cir.1996); Benjamin v. Fischer, 248 F.Supp.2d 251, 261-62 (S.D.N.Y. 2002); United States v. Pinto, 486 F.Supp. 578, 579, 582 (E.D.Pa.1980); see also United States v. Schwarz, 283 F.3d 76, 88-89 (2d Cir.2002) (district court admitted evidence of police officer's guilty plea, but did not admit details of his allocution, which constituted off-the-record evidence that the jury subsequently discovered). [9] Stewart argues that the Government’s theory and presentation cast the ImClone trade as the crime of insider trading. We concur with the District Court that the Government never crossed that threshold and pressed only that Stewart may have feared that she traded on inside information. [10] We are aware that Scop involved expert testimony regarding charged conduct (i.e., the ultimate legal question submitted to the jury), while here the government sought to preclude expert testimony regarding the legality of uncharged conduct (i.e., conduct that might create a motive to lie). However, at least in the circumstances of this case, we believe that this difference is immaterial because, in addition to invading the trial judge's exclusive authority to explain the law to the jury, expert testimony regarding an issue not relevant to the charges under consideration has the potential to confuse the jury even further. [11] Likewise, Defendants argue that the District Court improperly excluded argument and evidence (in addition to the securities law expert's testimony) that an ImClone “tip” would have been legal. For the same reasons that preclusion of the expert witness was appropriate, we conclude that the District Court's exclusion of the arguments relating to the legal status of insider trading was a proper exercise of its discretion. [12] Stewart indicates that she adopts and incorporates by reference, inter alia, Bacanovic's arguments regarding severance of the offense charged in Count Nine and regarding *314 the sufficiency of the evidence and the jury instruction of the offense charged in Count Two. It is unclear how these two arguments would apply to her. [13] United States v. Jones, 16 F.3d 487 (2d Cir.1994), is not to the contrary. In Jones, this Court indicated that under certain circumstances limiting instructions are insufficient to cure the risk of prejudice. See id. at 493. Here, unlike Jones, there was not an "overwhelming probability” that the jury would be unable to follow the District Court’s instructions, and the evidence was not "devastating to the defense.” Id. The District Court's limiting instructions were explicit and clear, and a substantial amount of corroborating evidence supported Pasternak's testimony. [14] Bacanovic’s contention that the message log did not contradict anything Bacanovic testified to before the SEC and thus did not provide sufficient corroboration of Arm *318 strong's testimony to satisfy the two-witness rule is also without merit. While the message may not have been independently sufficient evidence of Bacanovic's perjury, this Court’s precedents do not require that level of evidentiary value. See Weiner, 479 F.2d at 926-27.