Etingof's conjecture on the graded pieces of rational double affine Hecke algebra representations

Let $\Gamma_n$ be the object whose $(i,j)$-graded piece is denoted by ${\rm{gr}}_{i,j}(\Gamma_n)$. Fix a rational parameter $h$ with denominator $m>1$, let $\tilde{\mathfrak a}=\tilde{\mathfrak a}_\Lambda$ be the Lie subalgebra defined by the level-$1$ weight $\Lambda$, and let $P^{\tilde{\mathfrak a}}_+$ be its set of dominant integral weights. For $\mu\in P^{\tilde{\mathfrak a}}_+$, write $V^{\tilde{\mathfrak a}}_\mu$ for the corresponding irreducible integrable module, and let $V^{\tilde{\mathfrak a}}_\mu[\omega_0-n\delta,j]$ denote the subspace of weight $\omega_0-n\delta$ on which the $m$-th Casimir operator has eigenvalue $j$. Etingof's conjecture. There exists an isomorphism of $\mathbb{C}$-vector spaces

$${\rm{gr}}_{i,j}(\Gamma_n)=\bigoplus_\mu V^{\tilde{\mathfrak a}}_\mu[\omega_0-n\delta,j]\otimes {\rm{Hom}}_{\tilde{\mathfrak a}}(V^{\tilde{\mathfrak a}}_\mu,V_{\omega_0}^{\widetilde{\mathfrak{g}\mathfrak{l}}_\ell}),$$

where the sum is over all weights $\mu\in P^{\tilde{\mathfrak a}}_+$ such that $\langle\mu,\mu\rangle=-2i$. This conjecture predicts a representation-theoretic description of the bigraded pieces associated with the rational double affine Hecke algebra, relating them to modules for the subalgebra $\tilde{\mathfrak a}$.