Finkelberg–Mirković conjecture for the principal block

Let $G$ be as in the paper, let $\mathrm{Gr}^{\circ}$ be the connected component of the affine Grassmannian containing the base point, and let $I_{\mathrm{u}}$ be the preimage of the unipotent radical of $B$ under $\mathcal{L}^+G \to G$. Let $W_{\mathrm{aff}}^S$ denote the elements of the affine Weyl group $W_{\mathrm{aff}}$ that are minimal in their right cosets modulo the finite Weyl group $W$. Let $\widecheck{\mathbf{G}}$ be a connected reductive group whose Frobenius twist is $G^\vee_{\Bbbk}$, let $\mathsf{Rep}_0(\widecheck{\mathbf{G}})$ be its principal block, and write $\cdot_\ell$ for the dot action of $W_{\mathrm{aff}}$ on the character lattice. Finkelberg–Mirković conjecture. Assume that $\ell \geq h$. There exists an equivalence of categories

$$\mathsf{FM}: \mathsf{Perv}_{I_{\mathrm{u}}}(\mathrm{Gr}^\circ,\Bbbk) \xrightarrow{\sim} \mathsf{Rep}_0(\widecheck{\mathbf{G}})$$

which identifies the natural highest weight structures on both sides and sends, for every $w \in W_{\mathrm{aff}}^S$, the intersection cohomology complex of the orbit labelled by $w$ to the simple $\widecheck{\mathbf{G}}$-module of highest weight $w^{-1} \cdot_\ell 0$. Moreover, for $\mathcal{F} \in \mathsf{Perv}_{I_{\mathrm{u}}}(\mathrm{Gr}^\circ,\Bbbk)$ and $\mathcal{G} \in \mathsf{Perv}_{\mathcal{L}^+G}(\mathrm{Gr}^\circ,\Bbbk)$, there is a bifunctorial isomorphism

$$\mathsf{FM}(\mathcal{F} \star^{\mathcal{L}^+G} \mathcal{G}) \cong \mathsf{FM}(\mathcal{F}) \otimes \mathrm{Fr}^*\bigl(\mathsf{Sat}(\mathrm{sw}^* \mathcal{G})\bigr).$$

This is a central expected relationship between equivariant perverse sheaves on the affine Grassmannian and representations in the principal block; the supplied text gives no resolution status, so the conjecture is recorded as open.