Are there examples of conjectures that turned out to be false despite of strong heuristics?

There is a conjecture that the number of primes between $x$ and $x+y$, inclusive, is never more than the number of primes between $2$ and $2+y$, inclusive. It seems reasonable, since anyone can see that the primes thin out as you go up, and it has never been disproved, but forty years ago it was proved to be in conflict with another unproved conjecture that has even better heuristic evidence, so it's now believed to be false.

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