In a pair of filings Tuesday night, shortly after Ellison concluded her first day of testimony, attorneys made a set of motions requesting the authority to address these issues, noting that Judge Lewis Kaplan has previously asked for advance notification should the defense want to raise the “advice-of-counsel” argument.

Ellison ran Alameda, the crypto hedge fund founded by Bankman-Fried with close ties to his now-bankrupt FTX crypto exchange. She is his former paramour and the star witness in the criminal fraud case against him. On Tuesday, she testified that she was consulted on FTX’s investment portfolio (though she said her advice was ignored).

In one filing, the defense said it wants to question Ellison about her knowledge of whatever role FTX and Alameda lawyers might have played in setting policies, such as activating auto-deletion tools on the messaging app Signal.

The Department of Justice has already asked the judge to block Bankman-Fried’s defense team from raising the topic of the current valuation of FTX investments, reiterating this opposition in a filing this week addressing Anthropic AI specifically and saying it was immaterial to the question of whether Bankman-Fried misappropriated FTX customer funds for personal use. The artificial intelligence firm announced several fundraising efforts recently, raising hopes that FTX creditors may receive a larger payout should the bankrupt exchange sell its stake in the AI firm.

Read more: Star Witness Caroline Ellison Says Sam Bankman-Fried Directed Her to Commit Fraud

In the second filing, Bankman-Fried’s team argued that the Anthropic news “reflects … important context” about FTX’s expected value analysis, and they should be able to ask broader questions about FTX’s portfolio.

At the end of the court session on Tuesday, the attorneys and Judge Lewis Kaplan went back and forth over the question of whether these issues should be raised during the trial.

Ellison is also a personal investor in Anthropic, an assistant U.S. attorney said after the jury filed out.

“This witness who’s on the stand made a personal investment in Anthropic and has knowledge of the company’s investment in Anthropic, and so in the event that the Court deems this admissible, it might be an issue that we want to raise with her,” said the prosecutor, Danielle Sassoon. “We don’t think that this is a permissible topic of questioning, but if it is, we may want to ask her questions about it.”

Other outstanding issues include the question of whether the DOJ should be allowed to call a witness from Ukraine to testify remotely.

The judge said he did not want to allow any witness that would be providing cumulative testimony – meaning testimony that’s just reinforcing other witnesses’ – and, without making a ruling one way or another, suggested that the prosecutors “better consider alternatives.”

Another assistant U.S. attorney, Thane Rehn, said he would get back to the judge later this week.

Read all of CoinDesk’s coverage here.