Schur positivity conjecture for skew modified Hall–Littlewood functions

Let $\lambda$ and $\mu$ be partitions, and let $\widetilde{H}_{\lambda/\mu}(x;t)$ denote the skew modified Hall–Littlewood function defined by

$$\left\langle \widetilde{H}_{\lambda/\mu},\widetilde{P}_\nu\right\rangle:=\left\langle \widetilde{H}_\lambda,\widetilde{P}_\mu\widetilde{P}_\nu\right\rangle.$$

Writing its Schur expansion as

$$\widetilde{H}_{\lambda/\mu}(x;t)=\sum_\nu a^\nu_{\lambda\mu}(t)s_\nu,$$

Schur positivity conjecture. The coefficients $a^\nu_{\lambda\mu}(t)$ are nonnegative integer polynomials in $t$.

This conjecture asserts Schur positivity with coefficientwise positivity in the parameter $t$ for skew modified Hall–Littlewood functions; the supplied text does not indicate whether it has been proved or refuted.