Kostka-matrix termwise divisibility conjecture

Let $p$ be prime, let $e<f$ be positive integers, and set $m=p^e$, $n=p^f$, and $k=n/m=p^{f-e}$. Let $M^{se}$ and $M^{es}$ denote the change-of-basis matrices between the Schur and elementary bases, and let $\widetilde{\lambda}^*$ denote the transpose dual of $\lambda$. For every tuple $\{c_{\lambda\mu\nu}\}$ of nonnegative integers whose entries sum to $2(k-1)$, Kostka-matrix termwise divisibility conjecture. One has

$$p^{f-e}\mid \binom{2(k-1)}{\{c_{\lambda\mu\nu}\}}\left(\prod_{\lambda,\mu,\nu}M^{se}_{\lambda\mu}M^{se}_{\widetilde{\lambda}^*\nu}\right)M^{es}_{\sum_{\lambda,\mu,\nu}c_{\lambda\mu\nu}\mu,(n-m)^m}M^{es}_{\sum_{\lambda,\mu,\nu}c_{\lambda\mu\nu}\nu,(n-m)^m}.$$

This is a reformulation of the preceding termwise conjecture in terms of change-of-basis coefficients, which are expressible through Kostka numbers. It gives a combinatorial divisibility condition that is stronger than the valuation claim for $g(m,n)$; the source presents it as a reformulation and does not report a general proof.