Etingof's refined finite-dimensional representation-count conjecture

Let $H_{c,k}(\Gamma_n)$ be the symplectic reflection algebra in the non-integral rational-parameter setting, and let $\mathfrak a$, $P_+^{\mathfrak a}$, and $L_\mu[\omega_0-n\delta,0]$ have the meanings above.

Etingof's refined representation-count conjecture. The number of isomorphism classes of finite-dimensional irreducible representations of $H_{c,k}(\Gamma_n)$ is

$$\sum_{\mu\in P_+^{\mathfrak a}:\,\mu^2=0}\dim L_\mu[\omega_0-n\delta,0].$$

This is the finite-dimensional specialization of the refined two-index filtration conjecture, obtained because finite-dimensional representations have $j=0$. It is presented as a conjectural consequence and remains open in general.