Necessary condition for modular Hilbert-series changes of quasi-invariants

Let $Q_m(n)$ denote the space of quasi-invariant polynomials in $n$ variables with parameter $m$, and let $p$ be prime. Suppose the Hilbert series of $Q_m(n)$ over $\mathbb F_p$ differs from its Hilbert series over $\mathbb C$. Necessary-condition conjecture. There exist integers $a\ge 0$ and $k\ge 0$ such that

$$\frac{mn(n-2)+\binom n2}{n(n-2)k+\binom n2-1}\le p^a\le\frac{mn}{nk+1}.$$

In particular, if $p>mn$, the Hilbert series over $\mathbb F_p$ is the same as over $\mathbb C$. This conjecture is supported by computer calculations, especially for $n=3,4$, and is connected to the modular representation theory of $S_n$; its general validity remains open.