Monomial-form conjecture for the vectors $q(j)$

Let $r=s+p^t(p-1)d$, with $p mid d$, $s=b+c(p-1)$, and suppose that $2\leq b\leq p$ and $1\leq c\leq p-1$. Fix $a_p$ such that $1<\nu(a_p)<p-1$, and let $t>\nu(a_p)+c$. Suppose also that $s>2\nu(a_p)$, $1\leq m\leq \operatorname{min}\{\nu,c-1-\epsilon\}$, and $(m,\nu)\neq(\nu,\nu(a_p))$. Monomial-form conjecture. The monomials $q(j)$ belong to $V_r^{(m+1)}+\operatorname{Ker}(P)$ for every $c-m\leq j\leq c-1$. This is presented as a consequence of the matrix-form conjecture and would provide the required vanishing of these monomials in the quotient used to study the local constancy of the associated mod $p$ Galois representations; no resolution is given in the supplied text.