Washington (CNN) The Justice Department and the House have asked a federal judge to postpone the court hearing Thursday about impeachment witness Charles Kupperman's testimony.

The judge, Richard Leon of the US District Court in Washington, has not yet responded.

Thursday has been shaping up to be one of the most consequential days in court so far for the House impeachment inquiry and the White House's attempts to stonewall. Two hearings are scheduled that will test the White House's claim that its staffers are immune from testifying.

Leon had scheduled the hearing after Kupperman, a former deputy national security adviser, asked the court whether he needed to testify. The House has subpoenaed Kupperman but the White House had blocked him from testifying on Monday, claiming absolute immunity for White House officials in the impeachment probe.

The House and Justice Department agreed to seek the postponement, according to their filing Tuesday, because another court hearing they must attend is happening the same afternoon before a different judge. That case is about immunity from congressional testimony for another White House staffer, former White House counsel Don McGahn, as the House seeks his testimony in its impeachment probe.

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