Finkelberg–Mirković conjecture for Iwahori-constructible perverse sheaves

Let $\boldsymbol{G}^ atur$ be a complex connected reductive group Langlands-dual to $\boldsymbol{G}$, let $\beta$ be a field of coefficients, and let $\mathrm{Gr}=\boldsymbol{G}^\natur(\mathbb{C}((z)))/\boldsymbol{G}^\natur(\mathbb{C}[[z]])$ be its affine Grassmannian. Write $\mathsf{Perv}_{(\mathrm{Iw})}(\mathrm{Gr},\beta)$ for the category of Iwahori-constructible perverse sheaves, $\mathsf{Rep}_\varnothing(G)$ for the relevant representation category, $\mathcal{IC}_\lambda$ for the simple perverse sheaf attached to the Iwahori orbit indexed by $\lambda$, and $\operatorname{For}^{\dot G}_G$ for Frobenius pullback. Finkelberg–Mirković conjecture. There exists an equivalence of highest-weight categories

$$Q: \mathsf{Perv}_{(\mathrm{Iw})}(\mathrm{Gr},\beta) \xrightarrow{\sim} \mathsf{Rep}_\varnothing(G)$$

such that $Q(\mathcal{IC}_\lambda)\cong\mathsf{L}(w_\lambda\bullet0)$ for every $\lambda\in\mathbf{X}$, and such that, for every $\mathcal{F}\in\mathsf{Perv}_{(\mathrm{Iw})}(\mathrm{Gr},\beta)$ and $\mathcal{G}\in\mathsf{Perv}_{\mathrm{sph}}(\mathrm{Gr},\beta)$, there is a bifunctorial isomorphism

$$Q(\mathcal{F}\star\mathcal{G})\cong Q(\mathcal{F})\otimes\operatorname{For}^{\dot G}_G(\mathcal{S}(\mathcal{G})).$$

This conjecture relates the Iwahori-constructible geometric Satake category to representations in the principal block and remains open at the moment.