Dickson-image conjecture for the $k=n$ term when $m=n+1$

Let $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $Q=S/(x_1^{q^m},\ldots,x_n^{q^m})$, and let $G=Gl_n(\mathbb{F}_q)$. Assume $m=n+1$ and consider the $k=n$ term of the conjectured Hilbert series for $Q^G$. An invariant polynomial of $Q$ is an image of an invariant polynomial in $S$ if it is obtained from an element of $S^G$ under the quotient map $S\to Q$. Dickson-image conjecture. For arbitrary $n$ and $m=n+1$, one can find invariant polynomials corresponding to the $k=n$ term that are images of invariant polynomials in $S$. This predicts that the relevant graded piece is represented by reductions of Dickson invariants; the source provides no general construction or proof.