In 1976 with the drafting of the update of the National Emergency statute in the aftermath of the Watergate scandal and Nixon's attempt to absorb extraordinary Presidential powers, new termination laws were codified to allow Congress to reclaim their Article 1 powers assumed by the president ceded by Congress.
50 USC § 1622 contains those specifics which include termination of a Presidential declaration of a National Emergency through a nullification process if both Chambers concur with a SIMPLE
MAJORITY VOTE in both houses. If one Chamber votes to terminate, the other house is required to vote, and NO FILIBUSTER for the Senate.
The Speaker of the House will likely prepare the Joint Resolution soonest, and McConnell has no damn choice but to hold a vote in the Senate per the requirements of the Statute. If the Resolution gets to the Senate, it likely will have enough GOP Senators crossing the line easily. If McConnell gets smart of a sudden, he'll turn his caucus loose to vote on their own.
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50 U.S. Code § 1622 - National emergencies ~~