Monotonicity conjecture for generalized Kostka polynomials

Let $R=R(\eta,\gamma)$ be a dominant sequence of rectangles, and let $R'$ be obtained from $R$ by replacing a subsequence

$$((k^{\alpha_1}),\dots,(k^{\alpha_l}))$$

by

$$((k^{\beta_1}),\dots,(k^{\beta_l})),$$

where $|\alpha|=|\beta|$ and $\alpha^+$ dominates $\beta^+$ in the partition order. Let $K_{\lambda;R}(q)$ denote the associated generalized Kostka polynomial. Monotonicity conjecture. For every partition $\lambda$,

$$K_{\lambda;R}(q)\le K_{\lambda;R'}(q)$$

coefficientwise. Such inequalities are motivated by graded module epimorphisms induced by inclusions of nilpotent orbit closures; the source presents this strengthened monotonicity statement as conjectural.