TITLE: Counterexamples against all odds
QUESTION [38 upvotes]: 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.

REPLY [3 votes]: Let $\mathsf{A} = (A_1,\dots,A_m)$ be a tuple of $d \times d$ matrices. The joint spectral radius (JSR) of $\mathsf{A}$ is $\mathrm{JSR}(\mathsf{A}) := \lim_{n\to\infty} \sup_{i_1,\dots,i_n} \|A_{i_1} \dots A_{i_n}\|^{1/n}$, where $\|.\|$ is any norm on $\mathrm{Mat}(d\times d) = \mathbb{R}^{d^2}$.
The JSR was introduced by Rota and Strang in 1960. In the case of a single matrix ($m=1$), the JSR is equal to the spectral radius, that is, the biggest modulus of an eigenvalue of the matrix.
For equivalent definitions of the JSR, see e.g. Jungers' monograph.
The finiteness conjecture of Lagarias and Wang (1995) asserted that for any tuple $\mathsf{A} = (A_1,\dots,A_m)$, there is a product $A_{i_1} \dots A_{i_n}$ of some finite length $n$ whose spectral radius is exactly equal to $[\mathrm{JSR}(\mathsf{A})]^n$. This conjecture was disproved in 2001 by Bousch and Mairesse. More counterexamples were constructed later, e.g. here, here, and here.
However, it is conjectured (see Conjecture 8 by Maesumi) that if $m \ge 2$ and $d\ge 2$, then the counterexamples to the finiteness conjecture form a subset of $\mathbb{R}^{d^2m}$ of zero Lebesgue measure, so the finiteness conjecture is almost always true. This conjecture is supported by numerical evidence (see e.g. here), but so far remains entirely open.