Ren–Xu's conjecture on exceptional Hilbert series of quasi-invariants

Let $Q_m(n,\mathbf{F}_p)$ and $Q_m(n,\mathbf{Q})$ denote the corresponding quasi-invariant polynomial spaces. If their Hilbert series differ, then there exist integers $a\geq 0$ and $k\geq 0$ such that

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

Ren–Xu's conjecture. If the Hilbert series of $Q_m(n,\mathbf{F}_p)$ differs from that of $Q_m(n,\mathbf{Q})$, then there exist integers $a\geq 0$ and $k\geq 0$ satisfying the displayed double inequality. This conjecture gives a numerical criterion for the occurrence of Ren–Xu counterexamples, which explain why the Hilbert series in characteristic $p$ can differ from the characteristic-zero Hilbert series. The cited source attributes this conjecture to Ren and Xu; its resolution is not established by the supplied context.