Finkelberg–Mirković conjecture for the extended principal block

Let $G_{\Bbbk}$ be the semisimple group in characteristic $p$, let $\Repe$ be its extended principal block, let ${\EuScript Gr^\vee}$ be the affine Grassmannian of the Langlands dual group, and let $\Perv_{(\Iw)}({\EuScript Gr^\vee},k)$ be the category of $(\Iw)$-constructible perverse sheaves. Write $L_{x\pdot(-2\rho)}$ and $\Delta_{x\pdot(-2\rho)}$ for simple and standard objects, and $\ic_{x^{-1}}$ and $\Delta_{x^{-1}}$ for the corresponding perverse sheaves. Finkelberg–Mirković conjecture. There is an equivalence

$$\mathcal Q:\Repe\simeq\Perv_{(\Iw)}({\EuScript Gr^\vee},k)$$

that maps

$$L_{x\pdot(-2\rho)}\mapsto\ic_{x^{-1}},\qquad \Delta_{x\pdot(-2\rho)}\mapsto\Delta_{x^{-1}}$$

for every $x\in\Wae$ such that $x\pdot(-2\rho)\in X_+$, and is compatible with geometric Satake and Frobenius twist via the stated convolution diagram. The conjecture proposes a geometric realization of the extended principal block incorporating the linkage principle; the supplied text says it remains a conjecture, with progress by Achar, Riche, Achar–Rider, Mautner, and others.