The quasi-polynomiality conjecture for vanishing probabilities of Schur functions

Let $\lambda$ be a partition, and let $P(s_\lambda\mapsto0)$ denote the probability that the Schur function $s_\lambda$ evaluates to zero over the finite field under consideration. A function $g:\mathbb{Z}\to\mathbb{Z}$ is a quasi-polynomial if there exists an integer $N>0$ and polynomials $g_0,g_1,\ldots,g_{N-1}\in\mathbb{Z}[x]$ such that $g(n)=g_i(n)$ whenever $n\equiv i\pmod N$. The quasi-polynomiality conjecture. For every partition $\lambda$, there exist a positive integer $k$ and a quasi-polynomial $f(q)$, depending on the residue class of $q$ modulo some integer, such that

$$P(s_\lambda\mapsto0)=\frac{f(q)}{q^k}.$$

The claim predicts a uniform arithmetic form for these vanishing probabilities as the finite-field parameter $q$ varies; the examples preceding the statement exhibit such residue-class-dependent formulas, but no general proof is provided here.