TITLE: Have the tides ever turned twice on any open problem?
QUESTION [46 upvotes]: 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 $\mathsf{AM}$, but are not known to have certificates in $\mathsf{NP}$, a reasonable conjecture was that $\mathsf{NP}\subset\mathsf{AM}$.  However, based on the conjectured existence of strong-enough pseudorandom number generators, a reasonable statement nowadays is that $\mathsf{NP}=\mathsf{AM}$, etc.
I learned from Andrew Booker that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)\in \mathbb{Z}^3$ 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.

REPLY [4 votes]: $P=$ Carleson’s theorem, from his Abel interview:

Abel first thought that he had solved the general quintic by radicals. Then he found a mistake and subsequently he proved that it was impossible to solve the quintic algebraically. The famous and notoriously difficult problem about the pointwise convergence almost everywhere of L2-functions, which Lusin formulated in 1913 and actually goes back to Fourier in 1807, was solved by you in the mid-1960s. We understand that the prehistory of that result was converse to that of Abel’s, in the sense that you first tried to disprove it. Could you comment on that story?

(etc.)