Additive-basis conjecture for parabolic invariants when $m=2$

Let $P_\alpha$ be the parabolic subgroup of $Gl_n(\mathbb{F}_q)$ associated with a composition $\alpha$ of $n$, and let

$$Q=\mathbb{F}_q[x_1,\ldots,x_n]/(x_1^{q^2},\ldots,x_n^{q^2}).$$

Let $a_n$, $b_{r,k}$, $c_{r,s,k}$, and $d_r$ be the families of invariant polynomials constructed in the source. Additive-basis conjecture. For arbitrary $n$ and fixed composition $\alpha$ of $n$, these polynomials provide an additive basis for $Q^{P_\alpha}$. This would prove the parabolic Hilbert-series conjecture in the case $m=2$; the source does not establish the asserted basis in general.