I think Cheney will be ousted any day now from her number three position and then be primaried out come the 2022 GOP Wyoming primary. If I was Liz, I'd run was an independent which has the potential to split the GOP vote and let the democrat win. This may also apply to the 9 other house members.
History, in 2010 the Tea Party in Alaska ran Joe Miller who beat Murkowski in the Republican Primary making Miller the GOP nominee for senate in that state. Murkowski ran as an independent write in candidate and beat both Miller and the democrat in the General election. As an write in candidate which makes her accomplishment all the more impressive. Cheney is no Murkowski and Wyoming isn't Alaska, Liz would probably have to gather enough signatures to get on the ballot. I doubt she'd would be able to accomplishing anything as a write in.
Having worked for Perot, running as an independent is very rough. Republicans and Democrats get automatic ballot access, all others, third parties, independents must jump through hoops gathering signatures which differ state by state. 50 different requirements and 50 different time limits. Then both Republicans and Democrats go over the signatures with a fine tuned comb trying to throw out as many as possible to avoid letting an independent or third party candidate gain ballot access. You need to gather at least twice as many signatures as any state law requires, sometimes three times.
Perot also had to file many law suites and spend millions just to get on the ballot in all 50 states. There's much more, but the two major parties don't like anyone challenging them.
It's high past time that we start electing Americans to congress and the presidency who put America first instead of their political party. For way too long we have been electing Republicans and Democrats who happen to be Americans instead of Americans who happen to be Republicans and Democrats.