The inertial-parameter conjecture for mgs representations

Let $F$ be the function field of a smooth projective curve over a finite field, let $G$ be a split semisimple group over $F$, and let $u$ be a place. For a local representation $\pi_u$, let $\sigma_{\pi_u}:\operatorname{Gal}(\overline{F_u}|F_u)\to\widehat{G}(\overline{\mathbb Q}_\ell)$ be its semisimple Langlands parameter, and let $I_{F_u}$ denote the inertia subgroup of $\operatorname{Gal}(\overline{F_u}|F_u)$. Inertial-parameter conjecture. If $\pi_u$ is mgs, then the image of $I_{F_u}$ under $\sigma_{\pi_u}$ is not contained in any proper parabolic subgroup of $\widehat{G}(\overline{\mathbb Q}_\ell)$. This is expected to generalize the corresponding irreducibility statement and is consistent, in the epipelagic case, with the cited conjectures of Reeder and Yu. The paper notes that it would imply temperedness at all unramified places for an automorphic representation that is mgs at one place.