Putman–Wieland conjecture on homology orbits of characteristic surface covers

Let $\Sigma$ be the surface under consideration, let $\widetilde{\Sigma}\to\Sigma$ be a finite characteristic cover, and let $v\in {\rm H}_1(\widetilde{\Sigma},\mathbb{Q})$ be a nonzero homology class. Putman–Wieland conjecture. The ${\rm Mod}(\Sigma)$-orbit of $v$ is infinite.

This conjecture is essentially equivalent to Ivanov's conjecture that the first virtual Betti number of the mapping class group vanishes. It remains open; Kazhdan's property (T) for ${\rm Mod}(\Sigma)$ is a stronger open condition.