The adelic Mumford–Tate conjecture for Hodge-maximal motives

Let $M$ be a motive over a number field $F$ with Betti realization $M_B$, Mumford–Tate group $G_{MT}$, and adelic Galois representation

$$\hat{\rho}:\operatorname{Gal}_F\to G_{MT}(\hat{\mathbb Z}).$$

Assume that the Betti realization $M_B$ is Hodge-maximal. Adelic Mumford–Tate conjecture. The image of $\hat{\rho}$ is an open subgroup of $G_{MT}(\hat{\mathbb Z})$ in the adelic topology. In particular, for all sufficiently large primes $\ell$, the map

$$\rho_\ell:\operatorname{Gal}_F\to G_{MT}(\mathbb Z_\ell)$$

is surjective. This is an adelic strengthening of the Mumford–Tate conjecture. The source states it as a conjectural extension suggested by Serre; its general status is not resolved.