The short answer is NO.....the Dems would never allow that to happen, as it would split their party in two (their coalition is already crumbling, with the "progressives" in charge for the past five years), guaranteeing a Republican win in a walk.
As history is our teacher, there has never been a Democrat president running for reelection that has had a primary challenge, and went on to win the general election. I don't think that's happened in nearly a century. If an incumbent president is so unpopular that he earns a challenge from his own party, it's highly unlikely that he can prevail against the opposition party.
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