Positive characteristic form of the Langlands correspondence

Let $G$ be a connected reductive group over $\mathbb{Q}$, split over a number field $F$, and let $K$ be a level of $G$. Fix a prime $p$, let $f$ be a mod $p$ Hecke eigenform of level $K$, and for every good finite place $v$ of $F$ let $s_{f,v}\in\hat{G}(\overline{\mathbb{F}}_p)$ be its $v$-Satake parameter. Let $\Sigma$ be the finite set of places that are bad with respect to $p$ and $K$. Positive characteristic form of the Langlands correspondence. There exists a continuous representation

$$\rho\colon \operatorname{Gal}(\overline{F}/F)\longrightarrow\hat{G}(\overline{\mathbb{F}}_p)$$

that is unramified outside $\Sigma$ and satisfies

$$\rho(\operatorname{Frob}_v)=s_{f,v}$$

for every $v\notin\Sigma$. This is the expected relation between mod $p$ Hecke eigenforms and Galois representations, generalizing the correspondence conjectured for automorphic forms; the existence of such representations in this generality is not established.