Putman–Wieland Prym-orbit conjecture

Let $S_g$ be a surface of genus $g\geq 2$, and let a Prym representation be the linear representation of $\operatorname{Mod}(S_g)$ on the first homology of a finite characteristic cover of $S_g$. Putman–Wieland Prym-orbit conjecture. Every nonzero orbit of every Prym representation of $\operatorname{Mod}(S_g)$ is infinite. The source says this is essentially equivalent to the Putman–Wieland conjecture on finite-index subgroups surjecting onto $\mathbb Z$; the general statement remains open.