Positive characteristic form of Buzzard–Gee's conjecture on Galois representations attached to Hecke eigenforms

Let $G$ be the reductive group and $L$ its reflex field from the setup, let $K$ be the level, let $p$ be the coefficient characteristic, and let $\Sigma_{K,p}$ contain the places where the level or $p$ causes ramification. For a mod $p$ Hecke eigenform $f$ on $G$ of level $K$, let $s_{f,v}\in\hat{G}(\overline{\mathbb{F}}_p)$ be its semisimple $v$-Satake parameter at each finite place $v\notin\Sigma_{K,p}$. Positive characteristic form of Buzzard–Gee's conjecture. There exists a continuous representation

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

unramified outside $\Sigma_{K,p}$, such that for all $v\notin\Sigma_{K,p}$,

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

This predicts that the unramified Hecke data of every mod $p$ Hecke eigenform on the Shimura variety is encoded by a global mod $p$ Galois representation valued in the dual group. The supplied text does not state a resolution, so the conjecture is recorded as open.