Etingof's positivity conjecture for the finite-dimensional inner product

Let $\Gamma_n=S_n\ltimes\Gamma^n$ and let $H_{c,k}(\Gamma_n)$ be the associated symplectic reflection algebra. Assume that it has finitely many irreducible finite-dimensional modules, and write $K_0(H_{c,k}(\Gamma_n)-\operatorname{mod}_f)$ for the Grothendieck group of finite-dimensional modules. For finite-dimensional modules $M,N$, define

$$(M,N)=\dim \operatorname{RHom}(M,N),$$

where the dimension is taken in the supersense.

Etingof's positivity conjecture. The inner product $(,)$ is symmetric and positive definite, and in particular nondegenerate.

This conjecture concerns the categorical inner product on finite-dimensional representations of symplectic reflection algebras. The paper proves it for cyclic $\Gamma$ and arbitrary $n,c,k$; the general case is left open.