He will try but even the Supreme Court has shown they are willing to place limits on his power, ruling against him in some cases. So I don't think he has a pathway to bypassing the 22nd Amendment.

At the end of the day, the law is what people agree it is… And people don’t seem to be too interested in keeping the lunatic in check.

Putin would hope so because there's no other way.

He does not seem to care much for laws in general. Why change them when you can just ignore them? Much less work.