The hypothetical geometric Satake correspondence for the $B_{dR}$-affine Grassmannian

Let $G$ be a quasi-split reductive group over $E$, let $\Gamma=\operatorname{Gal}(\overline E/E)$, and let $L^+G$ act on the affine Grassmannian $\operatorname{Gr}$. Write ${}^LG=\widehat G\rtimes\Gamma$. Hypothetical geometric Satake correspondence. There should be a category of $L^+G$-equivariant $\overline{\mathbb{Q}}_\ell$-perverse sheaves on $\operatorname{Gr}$ equivalent to $\operatorname{Rep}_{\overline{\mathbb{Q}}_\ell}({}^LG)$, with the stated compatibility sending $IC_\mu$ to

$$r_\mu=\operatorname{Ind}_{\widehat G\rtimes\Gamma'}^{{}^LG}r_{\mu'}$$

when $\mu'\in\mu$ has stabilizer $\Gamma'$, and with $IC_\mu=\overline{\mathbb{Q}}_\ell(\langle\rho,\mu\rangle)[\langle2\rho,\mu\rangle]$ for minuscule $\mu$. This is the conjectural geometric Satake input needed to define the dual group intrinsically in the paper.