Conjecture on the partition-function bound in Harada's conjecture

For a positive integer $n$, let $p(n)$ be the number of partitions of $n$, and let $\Psi(\lambda)$ be the partition statistic used in the paper. Then

Partition-function bound conjecture. For $n\geq 7$,

$$p(n)\binom{n}{2}-\sum_{\lambda\vdash n}\Psi(\lambda)>0.$$

This inequality arises from the contribution of degree-one polynomials to the valuation polynomial associated with $h(\operatorname{GL}_n(k))$. Its validity would support the expected positivity behind the integrality of the Harada invariant in higher rank, but the conjecture is left open in the supplied text.