Hall–Littlewood positivity conjecture for orbit harmonics

Let $Z\subset\mathbb{C}^n$ be any finite $S_n$-stable set such that $\mathbb{C}[x_1,\dots,x_n]/\operatorname{gr}\mathbf{I}(Z)$ is a graded $S_n$-module, where $\mathbf{I}(Z)$ is the vanishing ideal of $Z$. Let $\operatorname{grFrob}(-;q)$ denote the graded Frobenius image, and let $\{\widetilde{H}_\lambda(\mathbf{x};q)\}$ be the Hall–Littlewood basis. Hall–Littlewood positivity conjecture. The expansion of

$$\operatorname{grFrob}(\mathbb{C}[x_1,\dots,x_n]/\operatorname{gr}\mathbf{I}(Z);q)$$

in the Hall–Littlewood basis has coefficients in $\mathbb{Z}_{\geq0}[q]$. This conjecture predicts positivity for graded orbit-harmonics modules, extending the examples discussed in the source. Its resolution status is not specified.