Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands correspondence constructs a geometrically irreducible Hecke eigensheaf on $Bun_2(C)$ associated to $\rho$, which is a perverse sheaf.

What is this sheaf's generic rank (as a function of $g$, presumably)? What is its singular locus?

It seems to me that the singular locus should be more-or-less independent of $\rho$ because in the complex-analytic picture it is supposed to depend smoothly on $\rho$, but the coordinates of $\rho$ are $\ell$-adic and the coordinates of the singular locus are in $\overline{\mathbb F}_q$ so there is no natural way for the second to depend naturally on the first other than being locally constant. If the singular locus is locally constant, because the complex-analytic space of local systems is connected, it should be globally constant as well. By similar logic it seems like the generic rank should be independent of the local system.

I could come up with some plausible guess for what the singular locus should be (maybe the unstable locus?), but I have no idea what the generic rank should be (some polynomial in $g$?).