Hodge-theoretic Koszul duality conjecture for Harish-Chandra sheaves

Let $G_{\mathbb R}$ be a real group with Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$, let $\mathcal M_{\rm Ho}$ be the full subcategory of $K$-equivariant Hodge $D$-modules whose underlying $D$-module belongs to $\mathcal M$, and let $\widetilde{\gr}$ denote the associated graded functor. Write $\widetilde{\mathfrak p}$ for the inverse image of $\mathfrak p$ in the universal conical variety, and let $\operatorname{Forg}$ forget the Hodge structure. Hodge-theoretic Koszul duality conjecture. There exists a full subcategory $\mathcal M^{\rm mix}\subset\mathcal M_{\rm Ho}$ that is a graded version of $\mathcal M$. For $M,N\in\mathcal M^{\rm mix}$, there is an isomorphism

$$\operatorname{Ext}^i(\operatorname{Forg}(M),\operatorname{Forg}(N))\cong \operatorname{Ext}^i_{\operatorname{Coh}^{K}(\widetilde{\mathfrak p})}(\gr(M),\gr(N))$$

for all $i$. The conjecture proposes that the Hodge-theoretic grading realizes the Ext-groups of the category of Harish-Chandra sheaves as coherent-sheaf Ext-groups; it is stated as open here, although the special case in which $G_{\mathbb R}$ is a complex group is said to be established elsewhere.