Etingof–Rains conjecture on Hilbert series of irreducible Cherednik-algebra quotients

Let $n=kp+r$ with $0\le r<p$. Define

$$[k]_z=\frac{1-z^k}{1-z},\qquad [k]_z!=[k]_z[k-1]_z\dotsm [1]_z,$$

and

$$Q_r(n,z)=\binom{n-1}{r-1}z^{r+1}+\sum_{i=0}^{r}\binom{n-r-2+i}{i}z^i.$$

Here $\mathcal{L}_{t,c}$ is the irreducible quotient of the polynomial representation of the rational Cherednik algebra, and $c$ is generic. Etingof–Rains conjecture. The Hilbert series of $\mathcal{L}_{t,c}$ has the form

$$h_{\mathcal{L}_{0,c}}(z)=[r]_z![p]_zQ_r(n,z)$$

and

$$h_{\mathcal{L}_{1,c}}(z)=[p]_z^{n-1}[r]_{z^p}![p]_{z^p}!Q_r\left(n,z^p\right).$$

This conjecture gives the proposed Hilbert-series formula in the general case $n=kp+r$ and is reported as unpublished. The formula holds in the cases established in the source and in the earlier work of Devadas and Sun, while the general case remains open.