Adelic commutator openness conjecture

Let $X$ be as above, let $H$ be the Mumford–Tate group, and let $H^{\operatorname{sc}}$ be the simply connected cover of its derived group. Let $\Phi_{\mathbb{A}}:G_K\to H(\mathbb{A}_f)$ be the adelic Galois representation, and let $\kappa:H\times H\to H^{\operatorname{sc}}$ be the lift of the commutator map. Adelic commutator openness conjecture. The subgroup generated by

$$\kappa(\Phi_{\mathbb{A}}(G_K),\Phi_{\mathbb{A}}(G_K))$$

is open in $H^{\operatorname{sc}}(\mathbb{A}_f)$. This is a weaker adelic openness statement designed to avoid the multiplicative part of the Galois image; the paper formulates it conditionally on the Mumford–Tate conjecture.