The motivic analogue of the Tate conjecture

Assume that the field $k$ is finitely generated, and let $M$ be a motive over $k$. Write $\mathop{\mathrm{G}}_{\ell,k}(M)$ for the $\ell$-adic monodromy group of $M$, and $\mathop{\mathrm{G}}_k(M)$ for its motivic Tannaka group. The notation $\mathop{\mathrm{G}}_k(M)_{\mathbb{Q}_\ell}$ denotes base change to $\mathbb{Q}_\ell$. Motivic Tate conjecture. The equality

$$\mathop{\mathrm{G}}_{\ell,k}(M)=\mathop{\mathrm{G}}_k(M)_{\mathbb{Q}_\ell}$$

should hold for every prime $\ell$. This is stronger than the classical $\ell$-adic Tate conjecture for a smooth projective variety, since it asserts that all Tate classes in all cohomology groups of all powers of the variety are motivated cycle classes.