Termwise divisibility conjecture for the symmetric-function expansion of g(m,n)

Let $p$ be prime, let $e<f$ be positive integers, and set $m=p^e$, $n=p^f$ as above. With $n=km$, consider the symmetric-function expression

$$\sum_{\lambda\subset (n-m)^m}s_{\lambda}s'_{\widetilde{\lambda}^*},$$

where the sum runs over partitions fitting in an $m$ by $n-m$ rectangle, and $s'_{\widetilde{\lambda}^*}$ denotes the Schur polynomial in the second set of variables associated with the transpose dual. Termwise divisibility conjecture. After expanding this expression in the elementary basis and taking the multinomial expansion of its $2(p^{f-e}-1)$-st power, every term paired with $s_{(n-m)^m}s'_{(n-m)^m}$ is divisible by $p^{f-e}$.

This is a stronger termwise formulation that would imply the lower-bound direction of the prime-power divisibility conjecture for $g(m,n)$. It arises from rewriting the relevant Schur inner product using the Cauchy identity; the source proves the original conjecture in special cases, but gives no resolution of this stronger termwise statement.