The unramified relative local Langlands conjecture for polarized hyperspherical varieties

Let $G$ be a complex reductive group, let $X$ be a smooth affine spherical $G$-variety with connected stabilizers in the open Borel orbit, and set $M=T^*(2)(X)$. Let $(\check{G},\check{M})$ be the corresponding hyperspherical dual, and let $L^+G$, $LX$, and $\operatorname{Gr}$ denote the positive loop group, loop space, and affine Grassmannian. Write $\cD_c(-)$ for the stable $\infty$-category of coherent $\cD$-modules, and let $\cD_c(L^+G\backslash LX)^{\Sat}$ be the full subcategory generated under convolution by the unramified basic object $\operatorname{IC}_0$, the $!$-extension of the constant $\cD$-module on $L^+X$. Unramified relative local Langlands conjecture. There is an equivalence of categories

$$\cD_c(L^+G\backslash LX)^{\Sat}\simeq \operatorname{Perf}(\sh^{1/2}(\check{M})/\check{G})$$

such that it is compatible with the action of derived geometric Satake,

$$\cD_c(L^+G\backslash LG/L^+G)\simeq \operatorname{Perf}(\check{\mathfrak{g}}^*[2]/\check{G}),$$

and sends $\operatorname{IC}_0$ to $\mathcal{O}_{\sh^{1/2}(\check{M})}$. This is a categorical form of the unramified relative local Langlands program, relating convolution categories of coherent $\mathcal{D}$-modules on loop spaces to perfect complexes on the dual Hamiltonian space. The source presents it as a conjecture and gives no resolution evidence.