Higher-order Macdonald positivity conjecture

Let $\lambda^1,\dots,\lambda^r$ be partitions, and define the Macdonald cumulant by

$$\kappa(\lambda^1,\dots,\lambda^r)(\bm{x};q,t):=\frac{\kappa(\tilde{H}_{\lambda^1}(\bm{x};q,t),\dots,\tilde{H}_{\lambda^r}(\bm{x};q,t))}{(q-1)^{r-1}}.$$

Writing it in the Schur basis defines the multivariate $q,t$-Kostka numbers by

$$\kappa(\lambda^1,\dots,\lambda^r):=\sum_{\mu}\tilde{K}^{(q,t)}_{\mu;\lambda^1,\dots,\lambda^r}\,s_\mu.$$

Higher-order Macdonald positivity conjecture. For every partition $\mu$, the multivariate $q,t$-Kostka number $\tilde{K}^{(q,t)}_{\mu;\lambda^1,\dots,\lambda^r}$ is a polynomial in $q,t$ with nonnegative integer coefficients.

The conjecture strengthens the known polynomiality of Macdonald cumulants to Schur-positivity. The source says it was motivated by extensive computer simulations and remained unresolved there.