The independence-of-ll conjecture for arithmetic monodromy groups and their representations

Let $X$ be a smooth curve over a finite field of characteristic $p$, let $E$ be a number field, and let ${\mathbf{L}}=\{{\mathcal{L}}_\lambda\}$ be an $E$-compatible system of lisse sheaves on $X$. For a geometric point ${\bar{\eta}}$ of $X$, write ${\operatorname{G_{arith}}}({\mathcal{L}}_\lambda,{\bar{\eta}})$ for the arithmetic monodromy group and let $\sigma_\lambda$ be its tautological representation. Assume that ${\mathbf{L}}$ is semisimple and pure of weight $w$ for some integer $w$. Independence-of-$\ell$ conjecture. (i) There exist a finite extension $F$ of $E$ and an algebraic group $G$ over $F$ such that, for every place $\lambda\in|F|_{\ne p}$, writing $\lambda$ also for its restriction to $E$, the $F_\lambda$-groups

$$G\otimes_F F_\lambda \qquad\text{and}\qquad {\operatorname{G_{arith}}}({\mathcal{L}}_\lambda,{\bar{\eta}})\otimes_{E_\lambda}F_\lambda$$

are isomorphic. (ii) Assuming (i), after replacing $F$ by a further finite extension, there exists an $F$-rational representation $\sigma$ of $G$ such that, for every such place $\lambda$, and after identifying the two $F_\lambda$-groups via an isomorphism from (i), the representations

$$\sigma\otimes_FF_\lambda \qquad\text{and}\qquad \sigma_\lambda\otimes_{E_\lambda}F_\lambda$$

are isomorphic. The conjecture asserts that the arithmetic monodromy groups, together with their tautological representations, arise by extension of scalars from common objects over a number field. Its motivation is the expected independence of $\ell$ for motivic compatible systems; the paper addresses this conjecture, but the supplied text gives no resolution, so its status is recorded as open.