“the chief justice could get 4 other votes pretty easily if he asked scotus to take it up.”
There’s no way this Chief Justice would want to take up a case like this. Also I don’t see a Chief Justice easily getting four members to agree with him on anything. There is no “easy” when it comes to the SC.
Dershowitz says that under Marbury/Madison the Scotus is the arbiter of anything between Congress/President.
A falsified or inadequate impeachment charge would definitely get the attention of Alito, Thomas, Kavenaugh, and Gorsuch.
It is entirely possible Roberts is a never-Trumper republican, but even he should see that the precedent for a political impeachment would mean every time an opposite party House gets in power that they would move to impeach a president over political differences.