The motivic Sato–Tate conjecture

Let $M$ be a motive and let $\operatorname{MS}_{\sim}(M)$ be its motivic Serre group. Motivic Sato–Tate conjecture. For any prime number $l$,

$$G_{l,K,1}^{\operatorname{alg}}=\operatorname{MS}_{\sim}(M)_{\mathbb Q_l}.$$

The paper obtains this formulation as equivalent to the motivic Mumford–Tate conjecture; it remains open in general.