Fixed-core generating function conjecture for hook-difference statistics

Let $\lambda_m$ be an $m$-core, and let $\mathcal{P}_{\lambda_m}$ be the set of partitions whose $m$-core is $\lambda_m$. For non-negative integers $\alpha$ and $\beta$ with $\alpha+\beta=m$, let $h_{\alpha,\beta}(\mu)$ denote the associated hook-difference statistic.

Fixed-core generating function conjecture. One has

$$\sum_{\mu\in\mathcal{P}_{\lambda_m}}t^{h_{\alpha,\beta}(\mu)}q^{|\mu|}=\frac{q^{|\lambda_m|}}{\prod_{i\geq 1}(1-q^{mi})^{m-1}(1-tq^{mi})}.$$

This generalizes the preceding fixed-2-core conjecture. The paper presents it as a conjectural extension of known generating-function identities; its validity for arbitrary $m$-cores and the stated parameters is left open.