Compatibility conjecture for the associated graded and block variety functors

Let $\mathcal M^{\rm mix}$ be the graded subcategory from the Hodge-theoretic Koszul duality conjecture. Let $\kappa$ be the functor from $K$-equivariant coherent sheaves on $\widetilde{\mathfrak p}$ to coherent sheaves on the corresponding character-space fiber product, and let $\widetilde T$ be the functor from $\mathcal M$ to coherent sheaves on the block variety. Compatibility conjecture. For $M,N\in\mathcal M^{\rm mix}$, there is a canonical isomorphism

$$\kappa\circ\gr(M)\cong \widetilde T(\operatorname{Forg}(M)).$$

This asserts compatibility between the associated graded construction for Hodge $D$-modules and the block-variety functor. The supplied text gives no resolution status or further evidence beyond stating the conjecture.