Background
Recently I read about some attempts of finding the exact order of Mertens function. It is conjectured that $$0<\limsup_{x\to\infty}\frac{M(x)}{\sqrt x(\log\log\log x)^{5/4}}<\infty\
(\text{Gonek's conjecture}),$$ and this conjecture is supported by probabilistic evidence that $M(e^y)e^{-y/2}$ has a limiting distribution. It would be surprising either if this conjecture is proven (which leads to Riemann Hypothesis) or if it is proven false despite of the strong heuristic probabilistic evidence.
Question: is there any famous conjecture which has probabilistic evidence to suggest it is true initially, but later it was proven false?
By probabilistic evidence I mean evidence such as $S=\sum_{n=1}^\infty\frac{\sin n^2x}n$ converges with probability one, suggesting that series $\sum_{n=1}^\infty\frac{\sin n^2}n$ converges. Note that it does not mean for all $x$, $S$ converges, but such an $x$ making $S$ diverges is hard to construct so even if there is an $x$ satisfying that condition, it does not count as an example.
Another possible example will be the normality of $\sqrt2$ in a chosen base. Borel proved$^{[1]}$ that almost all (w.r.t. Lebesgue measure) real numbers are normal. This is still open, and I wonder whether there is an irrational number which is proven not to be normal, but is not constructed digit-by-digit such as $0.123456789101112131415161718\cdots$. Here the probabilistic evidence is the result Borel proved.
[1]Borel, É. "Les probabilités dénombrables et leurs applications arithmétiques." Rend. Circ. Mat. Palermo 27, 247-271, 1909.