Finkelberg–Gaitsgory–Travkin $\bZ/2$-graded Koszul duality conjecture

Let $B\subset G$ and $B^{\vee}\subset G^{\vee}$ be Borel subgroups, and let $D^{\mathbb Z/2}$ denote the $\mathbb Z/2$-graded version of a graded category. $\mathbb Z/2$-graded Koszul duality conjecture. The $B$-equivariant category

$$(D(\fY)^{\mathbb Z/2})^B$$

is equivalent to

$$(Q_{\hbar=1}(\fX^{\vee})\operatorname{-mod}^{\mathbb Z/2})^{B^{\vee}_{\operatorname{mon}}}.$$

Similarly,

$$(D(\fY)^{\mathbb Z/2})^{B_{\operatorname{mon}}}$$

is equivalent to

$$(Q_{\hbar=1}(\fX^{\vee})\operatorname{-mod}^{\mathbb Z/2})^{B^{\vee}}.$$

This is the ungraded-parity formulation intended to avoid the additional grading issue in the preceding Koszul duality conjecture; the source gives no resolution status.