Vanishing of the zeroth cohomology for symmetric powers

Let $H_k^0$ denote the zeroth cohomology space associated with the $k$-th symmetric power, and let $M_k(T)$ be the corresponding polynomial or rational function defined from Frobenius characteristic polynomials.

Vanishing conjecture. $H_k^0 = 0$ for all positive integers $k$. Consequently, $M_k(T)$ is a polynomial for all positive integers $k$.

The paper proves this vanishing when $k$ is odd or when $k$ is even with $k<2p$. The conjecture concerns the remaining positive even values of $k$ and would imply polynomiality of $M_k(T)$ in every positive degree.