Gorenstein and palindromic Hilbert-series conjecture for Cherednik quotients

Let $M_c(S_n,\mathfrak{h},\tau)$ be the polynomial representation and $J_c(S_n,\mathfrak{h},\tau)$ its singular ideal, and write $L_c(S_n,\mathfrak{h},\tau)=M_c(S_n,\mathfrak{h},\tau)/J_c(S_n,\mathfrak{h},\tau)$. Here $n$ is arbitrary, $\tau$ is an $S_n$-representation, and $c$ may lie in or outside $\mathbf{F}_p$.

Gorenstein and palindromicity conjecture. If $\tau$ is trivial, then $M_c(S_n,\mathfrak{h},\tau)/J_c(S_n,\mathfrak{h},\tau)$ is Gorenstein for all $n$, both when $c\in\mathbf{F}_p$ and when $c\notin\mathbf{F}_p$. More generally, for every $\tau$, the Hilbert series $h_{L_c(S_n,\mathfrak{h},\tau)}(t)$ is palindromic.

The conjecture predicts a strong duality property for these finite-dimensional Cherednik quotients. The source does not provide a general proof or a resolution status beyond stating the conjecture.