The q-Doran conjecture

Let $a$ and $b$ be positive integers, set $n=ab$, let $c$ be a divisor of $n$ with $a\leq c\leq b$, and set $d=n/c$. Write $\preceq_s$ for Schur positivity. The q-Doran conjecture.

$$0\preceq_s\frac{H_c[H_d]-H_a[H_b]}{1-q}.$$

Equivalently, the Schur expansion of this divided difference has coefficients in $\mathbb{N}[q]$.

This conjecture extends Doran's generalized Foulkes conjecture from complete homogeneous symmetric functions to the Hall–Littlewood polynomials used in the paper. The source reports computational evidence and special-case results, but leaves the general assertion open.