Relative Langlands duality and Koszul duality

Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_\psi(Y)$ is polarized, so that $Q(X)=D_\psi(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then using a variant of the $S^1$-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_\psi(Y)\operatorname{-mod}^B)^{{\mathbb Z}/2}$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{B^\vee,\operatorname{mon}})^{{\mathbb Z}/2}$.