Explicit basis conjecture for parabolic invariants of truncated polynomial rings

Let ${\mathcal{Q}}_m(n)=\mathbb{F}_q[x_1,\ldots,x_n]/(x_1^{q^m},\ldots,x_n^{q^m})$, let $P(\alpha)$ be the parabolic subgroup associated to a composition $\alpha=(\alpha_1,\ldots,\alpha_\ell)$, and let $\delta_{a;b}$, $\Phi$, and $\Delta_s^m$ be the operators and Dickson-algebra subspaces defined in the source. For a weak composition $\beta\leq\alpha$ with $|\beta|\leq m$, set $B_i=\beta_1+\cdots+\beta_i$ and $B_0=0$, and choose

$$f_i\in\Phi^{B_{i-1}}\Delta_{\beta_i}^{m-B_{i-1}}\subseteq\Delta_{B_i}^{m-B_i}.$$

Explicit parabolic basis conjecture. The resulting elements

$$\delta_{B_1+1;m}^{\alpha_1-\beta_1}\bigg(f_1\delta_{B_2+1;m}^{\alpha_2-\beta_2}\big(\cdots f_{\ell-1}\delta_{B_\ell+1;m}^{\alpha_\ell-\beta_\ell}(f_\ell)\big)\bigg)$$

form the set $\mathcal{B}_m(\alpha)$, and $\mathcal{B}_m(\alpha)$ is an $\mathbb{F}_q$-basis of ${\mathcal{Q}}_m(n)^{P(\alpha)}$. This refines the parabolic Hilbert-series conjecture by proposing explicit basis elements, but the source supplies no resolution status.