General-monodromy Koszul duality conjecture for spherical varieties

Let $\sT\subset B$ be a Cartan subgroup, let $\lambda$ be a regular coweight of $G$, and let $\sT_{\lambda}$ be the subgroup of $\sT$ generated by $\exp(2\pi i\lambda)$. Let $B_{\lambda}$ be the centralizer of $\sT_{\lambda}$ in $B$, and let $(Q_{\hbar=1}(\fX^{\vee})\operatorname{-mod})^{B^{\vee}_{\operatorname{mon},\lambda}}$ denote the category of $\lambda$-monodromic modules over $Q_{\hbar=1}(\fX^{\vee})$. General-monodromy Koszul duality conjecture. Under the assumptions stated immediately before the claim, the categories

$$D(\fY^{\sT_{\lambda}})^{B_{\lambda}}$$

and

$$(Q_{\hbar=1}(\fX^{\vee})\operatorname{-mod})^{B^{\vee}_{\operatorname{mon},\lambda}}$$

are Koszul dual. The preceding theorem asserts the analogous equivalence in the $\mathbb Z/2$-graded setting; this conjectural statement concerns general, not necessarily unipotent, monodromy, and the source gives no resolution status.