The non-Hodge-maximal adelic Mumford–Tate conjecture

Let $M=H^w(X)$ arise from a smooth projective variety $X$ over a number field $F$, and fix an embedding $\sigma:F\hookrightarrow\mathbb C$. Let $G_{MT}$ be the Mumford–Tate group of $M_B$, and let $\pi:G'\to G_{MT}$ be the central isogeny constructed in the source. Assume the Mumford–Tate conjecture holds for $M$ with respect to $\sigma$ for every prime $\ell$. Non-Hodge-maximal adelic Mumford–Tate conjecture. There is a finite extension $E/F$ such that the image of

$$\hat{\rho}(\operatorname{Gal}_E)$$

is open in the image of

$$\pi:G'(\hat{\mathbb Z})\to G_{MT}(\hat{\mathbb Z}).$$

This formulation accounts for the possible failure of Hodge-maximality by replacing the full adelic Mumford–Tate group with the image of a central isogenous group. Its general status is not resolved.