Serre's maximal motive conjecture

Let $E$ be a motive over $K$, and let $G_{\mathcal{M}(E)}$ be its motivic Galois group. Assume that $G_{\mathcal{M}(E)}$ is connected, and let

$$\Phi_E:G_K\longrightarrow G_{\mathcal{M}(E)}(\mathbb{A}_f)$$

be the adelic Galois representation attached to the $\ell$-adic realizations of $E$. The motive $E$ is maximal if it does not factor through any non-trivial isogeny from a connected reductive group to $G_{\mathcal{M}(E)}$. Serre's maximal motive conjecture. The following properties are equivalent: $E$ is maximal, and $\operatorname{Im}(\Phi_E)$ is open in $G_{\mathcal{M}(E)}(\mathbb{A}_f)$. This conjecture characterizes maximal motives by adelic openness of their Galois images and is open in general.