Serre's domination by maximal motives conjecture

Let $E$ be a motive over $K$, and let $G_{\mathcal{M}(E)}$ denote its motivic Galois group. Serre's domination by maximal motives conjecture. After replacing $K$ by a finite extension, there exists a maximal motive $E'$ dominating $E$ such that

$$G_{\mathcal{M}(E')}\longrightarrow G_{\mathcal{M}(E)}$$

is a connected covering. This conjecture proposes that every motive becomes dominated by a maximal one after a finite base-field extension; its status is open in general.