What are some examples of conjectures proved to be true generically (i.e. there is a dense $G_{\delta}$ of objects that affirm the conjecture) but are nevertheless false?
Also, it would be cool to see examples where the conjecture was proved true with probability 1, but was nevertheless false.
Of course one can manufacture statements from the spaces themselves, but I'm mostly interested in actual conjectures from contemporary mathematics whose resolution exhibited this pattern.
I am curious about this situation because sometimes, although we cannot find them easily, the "counterexample space" for a given conjecture may be quite large, yet inaccessible due to the limitations of existing techniques. For example, Tsirelson's conjecture and the Connes Embedding Conjecture were recently proven false, and although we cannot yet concretely construct a counterexample I see no reason to believe that counterexamples will necessarily be terribly rare objects…once the techniques are available to construct them. (These may be fighting words.)
The present question inquires about a distinct situation where it has been proved that a randomly selected object will not provide a counterexample. As dire as this sounds, the situation may be advantageous in that the construction of a counterexample may have to be much more surgical, and so one may see more clearly a way to build one. I'm wondering if my intuition about this is valid, based on recent history.
The question is just a passing curiosity, really, but I think someone may have a good story or two that will educate.
Best Answer
The most famous example is the so-called Riemann-Hilbert problem, which has a long and complicated history which I don't explain in detail. As it happens Hilbert's own formulation was not very exact, this was rather a program of research than an exact formulation with a yes/no answer. This was Problem 21 in his famous list. Hilbert believed that the question has a positive answer, and even that he solved it.
The most common version of the problem was whether there exists a Fuchsian system, that is a differential equation of the form $$w'=A(z)w=\left(\sum_{j=1}^m\frac{A_j}{z-a_j}\right)w$$ on the Riemann sphere, with arbitrary prescribed singularities $a_j$ and prescribed monodromy representation. Here $A_j$ are constant $n\times n$ matrices, $w$ is a solution vector, and $w'=dw/dz$.
It was solved for $A$ in general position by Josip Plemelj in 1908, who obtained a positive answer, and for a long time it was assumed that the statement is true in general. It is true in dimension $2$, and it is true in higher dimensions under various very mild conditions which are violated on the set of large codimension. For example, the answer is positive if at least one $A_j$ is diagonalizable. However in 1989 Andrei Bolibrukh constructed a $3\times 3$ counterexample with $m=4$. Such counterexamples exist for every $n\geq 3$.
Ref. D. Anosov and A. Bolibruch, The Riemann-Hilbert problem
Bolibrukh, A. A. The Riemann-Hilbert problem on the complex projective line. (Russian) Mat. Zametki 46 (1989), no. 3, 118–120.