The base-changeability conjecture for mgs automorphic 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 $d70$ be an automorphic representation of $G(\mathbb A_F)$ that is mgs at a place $u$. A representation is base-changeable if there exists a finite set of mgs data at $u$ such that, for every constant field extension $F_n$ of $F$ of degree $n\geq 1$, there is a base change representation $\Pi_n$ of $G(\mathbb A_{F_n})$ which at places over $u$ is mgs with one of the given data, at the unramified places of $\pi$ is unramified and compatible under the Satake isomorphism, and at all other places has depth bounded independently of $n$. Base-changeability conjecture. Every automorphic representation of $G(\mathbb A_F)$ that is mgs at a place $u$ is base-changeable in this sense. This conjecture is introduced to support the transfer of automorphic representations through constant-field extensions and the geometric method used to establish temperedness; its resolution is not supplied here.