No. This is like the third time this cycle this rumor has been revived.
They'd be better off trying to switch Manchin.
Hm, this is the first time I heard about this rumor, but who knows, if the GOP has a Senate majority, King could be opportunistic enough to want to switch just because he can have more influence as a senator. I will concede that his conditions for switching seem like they would be difficult to fulfill. A senator doesn't just waltz in to become a committee chairman.
As for Manchin, he seems highly unlikely to switch parties anyway, so maybe this entire discussion just shows how unlikely it would be for a senator to leave one party for another these days.
