Mixed-parity Hecke and Satake formalism for two-fold Weil coverings

Assume the conjectural six-functor formalism for solid quasi-coherent sheaves over the $v$-topology. Let $R$ be a representation of the Langlands group $G^\mathrm{Lan}$ over the coefficient $A$, and let $f_A$ and $f_B$ be the Hecke-stack morphisms appearing in the source. The corresponding Hecke operation is

$$\mathrm{Hecke}(\square)=\mathrm{pushforward}_{f_B}\bigl(\mathrm{pullback}_{f_A}\square\otimes_{\blacksquare,\mathrm{complete}}\mathcal{F}_R\bigr).$$

Mixed-parity Hecke conjecture. In the mixed-parity setting this Hecke operator can be defined, its image lies in

$$D(\mathrm{Bundle}_{G,2})_{\mathrm{KL},\mathrm{lisse},\blacksquare,\widetilde{C}_\blacksquare},$$

and the Satake formalism can be proved so that the definition is well-defined. This would extend the local Hecke formalism to the $W_{K,2}$-equivariant setting, but the required six-functor and Satake foundations are assumed or conjectural in the source, so the claim remains open.