Mumford–Tate and Isogeny conjectures for Shimura-type structures

Let $(V_{\mathbb Z},\mu\circ s,\mu\circ\rho_{\widetilde{x},l})$ be an $(\infty,l)$-integral structure of Shimura type. Let $H_{\mathbb Q_l}$ be the identity component of the Zariski closure of the image of $\rho_{\widetilde{x},l}$, and let $M$ be the Mumford–Tate group of $s$. Mumford–Tate and Isogeny conjectures. The Mumford–Tate conjecture asserts

$$H_{\mathbb Q_l}=M_{\mathbb Q_l}.$$

The Isogeny conjecture asserts that every $(\infty,l)$-integral structure of Shimura type is Tate. These are the paper's combined formulations of the two conjectures; the source does not provide a resolution status for either assertion.