Filtration submodule and annihilation conjecture for general linear invariants

Let ${\mathcal{Q}}_m(n)=\mathbb{F}_q[x_1,\ldots,x_n]/(x_1^{q^m},\ldots,x_n^{q^m})$, let $G_n=GL_n(\mathbb{F}_q)$, let $\mathcal{A}$ be the mod-$q$ Steenrod algebra, and let ${\mathcal{D}}_n$ be the Dickson algebra with Dickson invariants $Q_{n,0},\ldots,Q_{n,n-1}$. For each $0\leq k\leq\min(m,n)$, let $\mathcal{F}_{n,k}$ be the span of $\delta_{s+1}^{n-s}(f)$ with $f\in\Delta_s^m$ and $0\leq s\leq k$.

Filtration submodule and annihilation conjecture. For each $1\leq k<\min(m,n)$, $\mathcal{F}_{n,k}$ is an $\mathcal{A}$-submodule and a ${\mathcal{D}}_n$-submodule of ${\mathcal{Q}}_m(n)^{G_n}$. Moreover, it is annihilated by

$$Q_{n,0},Q_{n,1},\ldots,Q_{n,n-k-1}.$$

The conjecture predicts additional Steenrod- and Dickson-module structure on the proposed invariant filtration. The source reports only that the filtration has these properties in the cases computed; no resolution status is supplied.