Etingof's refined filtration conjecture for non-integral rational parameters

Let $k$ be rational and non-integral with denominator $m>1$. Let $F_{i,j}$ and $\bold F_{i,j}$ be the two-index refinements of the geometric and support filtrations on $K_0(\operatorname{Rep}\Gamma_n)$, and define their associated graded pieces by

$$\operatorname{gr}_{i,j}^{F}K_0=F_{i,j}K_0/(F_{i-m,j+1}K_0+F_{i-1,j}K_0+F_{i,j-1}K_0),$$

with the analogous definition for $\operatorname{gr}_{i,j}^{\bold F}$. Let $\partial_m$ be the Heisenberg operator and let $L_\mu[\omega_0-n\delta,j]$ denote the subspace of weight $\omega_0-n\delta$ and $\partial_m$-eigenvalue $j$.

Etingof's refined filtration conjecture. For either filtration, there exists an isomorphism of vector spaces

$$\operatorname{gr}_{i,j}K_0(\operatorname{Rep}\Gamma_n)_{\mathbb C}\cong \bigoplus_{\mu\in P_+^{\mathfrak a}:\,\mu^2=-2i}L_\mu[\omega_0-n\delta,j]\otimes \operatorname{Hom}_{\mathfrak a}(L_\mu,\bold V).$$

This refines the one-index filtration conjecture by recording the Heisenberg degree. The paper notes that it subsumes the irrational case and gives the corresponding finite-dimensional representation count, but the general refined statement remains open.