Positivity conjecture for the cocharge-normalized generalized Kostka polynomial

For a sequence of rectangles $R$, let $r_{i,j}(R)$ be the number of rectangles containing $(i,j)$, and define

$$n(R)=\sum_{(i,j)}\binom{r_{i,j}(R)}{2}.$$

Set $\widetilde{K}_{\lambda;R}(q)=q^{n(R)}K_{\lambda;R}(q^{-1})$. Cocharge-normalized positivity conjecture. For $R$ dominant,

$$\widetilde{K}_{\lambda;R}(q)\in\mathbb{N}[q].$$

This is a reformulation of the positivity conjecture that also records the expected upper bound on powers of $q$.