Have the tides ever turned twice on any open problem?
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/\mathbb{Q}$, both the size of the torsion subgroup of $E(\mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: Conjecture There are only finitely many $E/\mathbb{Q}$ for which the rank of $E(\mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)
$P=$ Calabi’s conjecture.
Specifically, the link says “By the late 1960s, many were doubtful of the Calabi conjecture”, then Yau did “produce a "counterexample" to the conjecture. The "counterexample" was soon discovered to be flawed”, finally in 1976 Yau proved the conjecture. More details in Yau’s autobiography:
(pp. 76–77, 1971): Calabi had proposed a systematic strategy for constructing a vast number of manifolds endowed with special geometrical properties of which we’d never seen a single example. (...) Hitchin and I, along with many others, considered Calabi’s conjecture “too good to be true.”
(p. 86, 1972–73): with the French mathematician Jean-Pierre Bourguignon [we] tried out various approaches that might lead to the identification of a counterexample to the Calabi conjecture.
(p. 90, August 1973): I mentioned that I’d come up with a seemingly robust counterexample or two. Word got around, and I was asked to give an informal presentation (...) By the end of this session, most people left the room with the sense that I had proved Calabi wrong.
(p. 95, Fall 1973): Calabi (...) upon reflection had found some aspects of it puzzling. (...) When I went through the (...) counterexamples I’d been considering, one by one, they fell apart (...) So now I had to reverse my course 180 degrees and pour my efforts into proving that Calabi had been right all along.
(p. 104, Fall 1975): I was making steady progress (...) The proof, as I structured it, rested on four separate estimates to the critical complex Monge–Ampère equation.
(pp. 109–110, Fall 1976): I holed up in my study for as long as I could, pouring all of my energy into the Calabi conjecture. Within a week or two, the zeroth-order estimate had been completed and, consequently, the problem as a whole had been completed too.
I think the Busemann-Petty problem is an example like what you're asking, although changes in opinion would be due to progress (positive and negative) rather than any heuristic analysis.
A great description of the problem and what happened over time is the top answer at Widely accepted mathematical results that were later shown to be wrong?. See also https://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem.
It's a geometric conjecture (well, a "question" but I think it's easier to follow if I call it a conjecture) that is true in 2 dimensions and everyone expected it to hold in all higher dimensions, but things turned out differently with a surprise ending. Over time counterexamples were found to the conjecture in every dimension above 5, then it was proved to be true in dimension 3, then a counterexample was found in dimension 4, then the counterexample in dimension 4 was shown to be an example rather than a counterexample, and then it was proved to be true in dimension 4 by the same person who earlier gave the "counterexample" in dimension 4, with both of his papers appearing in the Annals.