Etingof's filtration conjecture for irrational parameters

Let $K_0(\operatorname{Rep}\Gamma_n)_{\mathbb C}$ be the complexified Grothendieck group, equipped with either filtration $F_\bullet$ or $\bold F_\bullet$. Let $\mathfrak a$ be the Lie subalgebra defined by the singular hyperplanes containing $(\lambda,k)$, let $P_+^{\mathfrak a}$ be its dominant integral weights, and let $L_\mu$ be the corresponding irreducible integrable $\mathfrak a$-module. Let $\bold V$ be the basic representation and $L_\mu[\omega_0-n\delta]$ its indicated weight space.

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

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

This is the main conjectural description of the filtration gradeds in the irrational-$k$ case. The paper derives vanishing consequences and verifies the prediction in some examples, but leaves the general assertion open.