The q-stability conjecture

For $0<a\leq b$, let $F_{a,b}(\mathbf{x};q)$ be the divided difference from the $q$-Foulkes conjecture, and let the overline operator remove the largest part from each Schur index. For a Schur function $s_\lambda$, the q-stability conjecture.

$$\left\langle\overline{F_{a,b+1}(\mathbf{x};q)}-\overline{F_{a,b}(\mathbf{x};q)},s_\lambda(\mathbf{x})\right\rangle\in\mathbb{N}[q].$$

This is the $q$-analogue of the stability part of Foulkes's conjecture. The paper presents it as a conjecture supported by examples; no general proof is given.