They say they expect that the newly appointed Trump-Russia special prosecutor, former FBI Director Robert Mueller, will—like Walsh 31 years ago—make an early request to Congress to hold off on immunity, even if that means curtailing the congressional inquiries dramatically by foregoing public hearings. The Republican leaders of House and Senate committees investigating the Trump-Russia issue have said they have no intention of offering immunity for now, but will they be tempted later if there is no other way to obtain information and the lawmakers are desperate to show progress?

The impasse could come sooner than Mueller might have hoped. A Washington criminal defense lawyer who is representing a key figure in the Trump-Russia investigation, speaking on condition of anonymity, told Politico Magazine last week that a number of witnesses connected to the Trump campaign who had previously offered to testify before Congress voluntarily, without precondition, are seriously considering whether to withdraw the offer and insist on immunity.

“With the appointment of Mueller, this investigation got a lot more real for everyone,” the lawyer said. “Why would you send clients up to Congress now and let them get caught up in a perjury trap over what could be a minor issue?” The lawyer would not identify which witnesses might withdraw their pledge to testify voluntarily, but among the Trump advisers who have previously offered to cooperate fully with Congress are Jared Kushner, the president’s son-in-law and a top White House adviser; former Trump campaign manager Paul Manafort; and campaign advisers Roger Stone and Carter Page.