Motivated Mumford--Tate conjecture

Let $k$ be a finitely generated subfield of $\mathbf C$, let $\ell$ be a prime, and let $M\in\operatorname{AM}(k)$. Let $r_{\mathrm B}(M)$ and $r_\ell(M)$ be its Betti and $\ell$-adic realizations, let $\gamma:r_{\mathrm B}(M)\otimes\mathbf Q_\ell\cong r_\ell(M)$ be the comparison isomorphism, and let $\mathcal G(r_\ell(M))$ be the Zariski closure of the Galois image. Motivated Mumford--Tate conjecture. Under $\gamma$,

$$\operatorname{MT}(r_{\mathrm B}(M))\otimes\mathbf Q_\ell=\operatorname{G_{mot}}(M_{\mathbf C})\otimes\mathbf Q_\ell\cong\operatorname{G_{mot}}_{,\ell}(M_{\overline k})=\mathcal G(r_\ell(M))^0.$$

The first equality is the motivated Hodge-class assertion, while the last is the analogous assertion for Tate classes; the conjecture strengthens the classical Mumford--Tate conjecture and remains open.