Integrality and nonnegativity of the hypergeometric coefficients $X^{(d)}_{m,h}(n;\mu)$

The quantities $X^{(d)}_{m,h}(n;\mu)$ are defined by the explicit formula preceding these observations, for the parameters and partition $\mu$ used there. Integrality conjecture. It holds that

$$X^{(d)}_{m,h}(n;\mu)\in\mathbb{Z}_{\ge 0}.$$

The conjecture is motivated by the tabulated values and, if true, suggests that these coefficients count a special type of cycle in the discrete torus $\mathrm{DT}^{(d)}_m$.