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If Kavanaugh were smart and ethical judge, he would recuse himself. But I think he is far more ambitious than ethical. He does not want sit on the sidelines during historical cases. If trump were to pardon himself, Kavanaugh would want to rule on it.
I don't like Kavanaugh and I hope something happens the causes his nomination to fail. However, I think it is doubtful that he and the rest of judges would allow trump pardon himself. The SC has not been waiting decades for someone like trump to become president just so they too can go down in history as being total morons that rule: "The president can commit all the crimes he wants and then pardon himself. Under the US Constitution, the president is above the law."
Realistically, trump would just make a private deal in advance with Pence before leaving office ensuring that Pence pardon's trump thus it would go unchallenged.
My prediction remains: Before Feb 2019, trump will suddenly pardon 100s of people including the entire WH staff, his family, many of his business partners who he knows have committed undiscovered crimes, his own business and other businesses, He may also include pardon's for Russia and Putin. THEN, he will resign that same day....
The SC will then be forced to rule if all his pardon are valid. Imagine the legal chaos if he did pardon Putin and Russia.