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 $X\simeq T_{\psi }^{\ast }(Y)$ 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 $\mathbb{Z}\mathrm{/}2$-graded $B$-equivariant category $(D_{\psi }(Y){\mathrm{-mod}}^B{)}^{\mathbb{Z}\mathrm{/}2}$ and the $\mathbb{Z}\mathrm{/}2$-graded unipotent $B^{\vee }$-monodromic category $(Q(X^{\vee }){\mathrm{-mod}}^{B^{\vee },\mathrm{mon}}{)}^{\mathbb{Z}\mathrm{/}2}$.