Bezrukavnikov–Gaitsgory–... relative Langlands duality conjecture for spherical varieties

Let $G$ be a connected reductive group over $\mathbb C$, let $\fY$ be a smooth affine spherical $G$-variety with connected stabilizer in a Borel subgroup $B$ of a general point, and let $\fX=T^*\fY$. Let $G^{\vee}$ be the Langlands dual group and $\fX^{\vee}$ the conjectural dual hyper-spherical affine Poisson $G^{\vee}$-variety, equipped with a graded quantization $Q_{\hbar}(\fX^{\vee})$. Write $\tilD(W)^A$ for the renormalized derived category of $A$-equivariant $D$-modules on $W$, and let $\CK$ and $\CO$ denote the Laurent and formal power series rings, respectively. Relative Langlands duality conjecture. The category

$$\tilD(\fY(\CK))^{G(\CO)\rtimes \mathbb C^\times}$$

is equivalent to

$$(Q_{\hbar}(\fX^{\vee})\operatorname{-mod})^{G^{\vee}}.$$

Here $\mathbb C^\times$ acts on the left by loop rotation, and its equivariant parameter corresponds to $\hbar$ on the right. This is one of the main conjectures formulated for dual hyper-spherical varieties; the source gives no resolution status.