Etingof's simplicity conjecture for symplectic reflection algebras

Let $\Gamma_n=S_n\ltimes\Gamma^n$, where $\Gamma\subset SL_2(\mathbb C)$, and let $H_{c,k}(\Gamma_n)$ be the corresponding symplectic reflection algebra. Write the parameters as $(\lambda,k)$, and let $\mathfrak g$ be the finite-dimensional Lie algebra associated with $\Gamma$. Define the hyperplanes

$$E_{m,N}: km+N=0,\qquad m\in\mathbb Z,\quad 2\le m\le n,\quad N\in\mathbb Z,\quad \operatorname{gcd}(m,N)=1,$$

and

$$H_{\alpha,m,N}: (\lambda,\alpha)+km+N=0,$$

where $|m|\le n-1$, $\alpha$ is a root of $\mathfrak g$, and $N\in\mathbb Z_{\ge 0}$.

Etingof's simplicity conjecture. The algebra $H_{c,k}(\Gamma_n)$ is simple if and only if $(\lambda,k)$ belongs to none of the hyperplanes $E_{m,N}$ or $H_{\alpha,m,N}$.

The conjecture makes precise the known statement that simplicity fails only on a countable collection of parameter hyperplanes. The paper states that it holds for $n=1$ and for all $n$ when $\Gamma$ is cyclic; the remaining cases are open.