Finkelberg–Gaitsgory–Travkin Koszul duality conjecture for spherical varieties

Let $G$ be a connected reductive group, let $\fY$ be the spherical variety above, and let $B\subset G$ and $B^{\vee}\subset G^{\vee}$ be Borel subgroups. Let $Q_{\hbar=1}(\fX^{\vee})$ be the ungraded specialization of the quantization, and let the subscript $\operatorname{mon}$ denote the derived category of modules with unipotent monodromy. Koszul duality conjecture. The categories

$$D(\fY)^B\quad\text{and}\quad (Q_{\hbar=1}(\fX^{\vee})\operatorname{-mod})^{B^{\vee}_{\operatorname{mon}}}$$

are Koszul dual. Similarly,

$$D(\fY)^{B_{\operatorname{mon}}}\quad\text{and}\quad (Q_{\hbar=1}(\fX^{\vee})\operatorname{-mod})^{B^{\vee}}$$

are Koszul dual. Koszul duality is understood to involve additional gradings; the source gives no resolution status.