Geometric Langlands in prime characteristic

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$, whose characteristic is positive and does not divide the order of the Weyl group of $G$, and let $\breve{G}$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$. Denote by ${\text{Bun}}_G$ the moduli stack of $G$-bundles on $C$ and ${\text{Loc}}_{\breve{G}}$ the moduli stack of $\breve{G}$-local systems on $C$. Let $D_{{\text{Bun}}_G}$ be the sheaf of crystalline differential operators on ${\text{Bun}}_G$. In this paper we construct an equivalence between the bounded derived category $D^b(\text{on}QCoh({\text{Loc}}_{\breve{G}}^0))$ of quasi-coherent sheaves on some open subset ${\text{Loc}}_{\breve{G}}^0\subset {\text{Loc}}_{\breve{G}}$ and bounded derived category $D^b(D_{{\text{Bun}}_G}^0\text{on}-mod)$ of modules over some localization $D_{{\text{Bun}}_G}^0$ of $D_{{\text{Bun}}_G}$. This generalizes the work of Bezrukavnikov-Braverman in the ${\text{GL}}_n$ case.