The derived Satake equivalence conjecture for spherical varieties

Let $F=\mathbb F_q((t))$, let $L$ and $L^+$ denote formal loop and arc spaces, and let $X$ and $G$ be defined over $\mathbb F_q$, with $X$ smooth. Let $\check M=V_X\times^{\check G_X}\check G$, where $V_X$ is the representation space of $\rho_X$, and let $\mathbb C[\check M]^{\mathbin{\mkern-6mu\fatslash}}$ be the sheared differential graded algebra introduced in the source. Derived Satake equivalence conjecture. After fixing an isomorphism $\mathbb C=\overline{\mathbb Q_\ell}$, there is an equivalence of triangulated $\mathbb C$-linear categories

$$\operatorname{Shv}(LX/L^+G)\xrightarrow{\sim}D_{\operatorname{per}}^{\mathbin{\mkern-6mu\fatslash}}(\check M/\check G),$$

where the category on the right is the full triangulated subcategory generated by perfect complexes in the category of $\check G$-equivariant differential graded $\mathbb C[\check M]^{\mathbin{\mkern-6mu\fatslash}}$-modules localized by quasi-isomorphisms. This is intended as a categorical form of the local $L$-value conjecture via the sheaf–function dictionary; it is presented as ongoing work and remains open.