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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 polynomial identity testing, have nice probabilistic solutions, i.e. they are in $mathsf{BPP}$, but are not known to have solutions in $mathsf{P}$, a reasonable conjecture was that $mathsf{P}subsetmathsf{BPP}$. Similarly it has been reasonable to suppose that $mathsf{NP}subsetmathsf{AM}$. However, based on the conjectured existence of strong-enough pseudorandom number generators, a reasonable statement nowadays is that $mathsf{P}=mathsf{BPP}$, etc.




• I learned from Andrew Booker that, after some computer experiments in the 1950's, the prevailing opinion was that for no $k$ congruent to $3$ or $6$ mod $9$ are there an infinite number of solutions to $x^3+y^3+z^3=k$ with $(x,y,z)in mathbb{Z}^3$. However, after some conjectured 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 more commonly known as the "second Hardy-Littlewood Conjecture". See also this MSF question. However, in 1974 Ian Richards showed that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood.

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 $E/mathbb{Q}$, 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 22. (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.