Conjectural Dickson-module generation for excellent polynomials

Fix $m\geq 0$. Let $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $A_m^{(n)}$ be the set of $(n+1,m)$-excellent polynomials, meaning polynomials $f$ for which $\delta_{n+1}(f)$ is a polynomial, and let $\mathcal{D}_n$ be the Dickson algebra. Write $\delta_n$ for $\delta_{n;m}$. Excellent-polynomial generation conjecture. For $n\geq 2$, $A_m^{(n)}$ is generated as a $\mathcal{D}_n$-module by

$$G_m^{(n)}=\{1\}\cup\{\delta_n(g)\mid g\in A_m^{(n-1)}\}.$$

The conjecture proposes an explicit generating set for the excellent-polynomial space as a module over the Dickson algebra, motivated by the source's partial results on polynomiality of the $\delta$ operators; no resolution is supplied.