A key hearing in the federal election interference case against former President Donald Trump got underway Thursday morning in Washington, D.C. This was the first in-court proceeding involving Trump since the U.S. Supreme Court's landmark decision on presidential immunity.

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Judge Tanya Chutkan will have to decide what evidence is allowed in this case.

This new court hearing comes two months after the Supreme Court's decision, with all six conservative justices ruling there is presidential immunity for official acts. The Supreme Court said presidents cannot be charged with crimes when actions involve official acts while in office.

The Supreme Court ruling put the brakes on four criminal cases against Trump.

Chutkan made clear that issues of presidential immunity needs to be "dealt with as early as possible."

The judge did not say when a potential trial could begin, but said, "The electoral process and the timing of the election is not relevant here. She added, This court is not concerned with the electoral schedule.

Legal experts said the Supreme Court's decision significantly weakened special counsel Jack Smith's case.

"The Supreme Court has left a lot of space now for the lower courts to try to interpret which of these actions count within the official scope of the president's actions and which are outside," said George Washington University law professor Paul Schiff Berman.

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Last week, Smith unveiled a new superseding indictment against the former president with the same criminal charges, including conspiracy and obstruction.

Prosecutors say Trump knowingly spread lies that there was fraud in the 2020 election. And on January 6, 2021, when rioters broke into the U.S. Capitol, he "exploited the disruption" for personal gain.

Trump has pleaded not guilty and calls this case "election interference," writing on Truth Social, "All of these scams will fail, just as deranged Jack's hoax in Florida has been fully dismissed, and we will win the most important election in the history of our country on November 5th."

A trial is not expected before the election.

"I think whatever Judge Chutkan does, it will almost certainly get appealed to the D.C. Circuit, and perhaps ultimately the Supreme Court will weigh in again," said Schiff Berman.

During Thursday's hearing, lawyers for the former president argued that conversations between Trump and Vice President Mike Pence should equate a dismissal. Chutkan rejected that argument.