Rodriguez-Villegas's integrality conjecture for structure coefficients

For partitions $\boldsymbol{\mu}$ and $\lambda$, let $c_{\boldsymbol{\mu}}^{\lambda}(q,t)$ denote the structure coefficients introduced through the product of modified Hall–Littlewood polynomials. Rodriguez-Villegas's integrality conjecture. The structure coefficients satisfy

$$c_{\boldsymbol{\mu}}^{\lambda}(q,t)\in\mathbb{Z}[q,t].$$

This conjecture concerns the positivity and integrality properties of the modified Kostka-polynomial structure constants; the source attributes it to unpublished notes by Rodriguez-Villegas, and no resolution is given.