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Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justification.

Some examples that come to mind:

• Because many decision problems, such as graph non-isomorphism, have nice probabilistic protocols, i.e. they are in $mathsfAM$, but are not known to have certificates in $mathsfNP$, a reasonable conjecture was that $mathsfNPsubsetmathsfAM$. However, based on the conjectured existence of strong-enough pseudorandom number generators, a reasonable statement nowadays is that $mathsfNP=mathsfAM$, etc.




• I learned from Andrew Booker that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)in mathbbZ^3$ have varied, especially after some heuristics from Heath-Brown. It is reasonable to state that most $k$ have an infinite number of solutions.




• Numerical evidence suggests that for all $x$, $y$, we have $pi(x+y)leq pi(x)+pi(y)$. This is commonly known as the "second Hardy-Littlewood Conjecture". See also this MSF question. However, a 1974 paper showed that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood.



• Number theory may also be littered with other such examples.

I'm interested if it has ever happened whether the process has ever repeated itself. That is:

Have there ever been situations wherein it is reasonable to suppose $P$, then, after some heuristic analysis, it is reasonable to supposed $neg P$, then, after further consideration, it is reasonable to suppose $P$?

I have read that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.

I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/mathbbQ$, 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 $E/mathbbQ$ for which the rank of $E(mathbbQ)$ 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: Yau’s autobiographical account.

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.

Hilbert's $16^rm th$ problem.

In 1923 Dulac "proved" that every polynomial vector field in the plane has finitely many cycles [D]. In 1955-57 Petrovskii and Landis "gave" bounds for the number of such cycles depending only on the degree of the polynomial [PL1], [PL2].

Coming from Hilbert, and being so central to Dynamical Systems developments, this work certainly "built a small industry". However, Novikov and Ilyashenko disproved [PL1] in the 60's, and later, in 1982, Ilyashenko found a serious gap in [D]. Thus, after 60 years the stat-of-the-art in that area was back almost to zero (except of course, people now had new tools and conjectures, and a better understanding of the problem!).

See Centennial History of Hilbert's 16th Problem (citations above are from there) which gives an excellent overview of the problem, its history, and what is currently known. In particular, the diagram in page 303 summarizes very well the ups and downs described above, and is a good candidate for a great mathematical figure.