Lewis–Reiner–Stanton parabolic Hilbert-series conjecture

Let $G=GL_n(\mathbb{F}_q)$ act on $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $Q(m,n)=S/(x_1^{q^m},\ldots,x_n^{q^m})$, and let $P_{\alpha}$ be the parabolic subgroup associated with a composition $\alpha=(\alpha_1,\ldots,\alpha_{\ell})$ of $n$. For $\beta\in\mathbb{Z}_{\geq 0}^{\ell}$, write $\beta\leq\alpha$ when $\beta_i\leq\alpha_i$ for every $i$, write $|\beta|=\sum_{j=1}^{\ell}\beta_j$, and set $B_i=\sum_{j=1}^{i}\beta_j$. Lewis–Reiner–Stanton's parabolic conjecture. For $m\geq 0$, the invariant ring $Q(m,n)^{P_{\alpha}}$ has Hilbert series

$$\operatorname{Hilb}(Q(m,n)^{P_{\alpha}},t)=\sum_{\substack{\beta\leq\alpha\beta\leq m}}t^{e(m,\alpha,\beta)}\genfrac{[}{]}{0pt}{}{m}{\beta,m-|\beta|}_{q,t},$$

where

$$e(m,\alpha,\beta)=\sum_{i=1}^{\ell}(\alpha_i-\beta_i)(q^m-q^{B_i}).$$

This conjecture predicts the Hilbert series of invariant rings of truncated polynomial rings under parabolic subgroups; the supplied source does not indicate whether it has been resolved.