Matrix-form conjecture on solvability of the systems associated with $A$

Let $r=s+p^t(p-1)d$, with $p mid d$, $s=b+c(p-1)$, and suppose that $2leq bleq p$, $1leq cleq p-1$, and $tgeq 2$. Let

$$A=\left(\alpha(i,l)\right)_{\substack{1\leq i\leq m+1\\0\leq l\leq m}},$$

where the coefficients $\alpha(i,l)$ are those defined in the source. Suppose also that $1\leq m\leq c-1-\epsilon$. Matrix-form conjecture. The linear systems $AX=e_i\pmod p$ have a solution for every $1\leq i\leq m$. This conjecture is motivated by SageMath computations and is used to deduce vanishing statements for the monomials $q(j)$ in the relevant quotient of the representation space; its status is not resolved in the supplied text.