The second Tate conjecture for log mixed motives

Let $S$ be of finite type over $\mathbb{Q}$, and let $M,N$ be objects of $\mathrm{LMM}_{\{\ell,B\}}(S)$. Write $M_{\ell},N_{\ell}$ for their $\ell$-adic realizations and $M_B,N_B$ for their Betti realizations on $(S\otimes\mathbb{C})_{\mathrm{an}}^{\log}$.

Second Tate conjecture. The set $\operatorname{Hom}(M,N)$ is in bijection with the set of pairs $(a,b)$ where $a:M_{\ell}\to N_{\ell}$ is a morphism and $b:M_B\to N_B$ is a homomorphism on $(S\otimes\mathbb{C})_{\mathrm{an}}^{\log}$ such that the pullback of $a$ to $(S\otimes\mathbb{C})_{\mathrm{an}}^{\log}$ is induced by $b$.

This conjecture is a compatibility and full-faithfulness assertion combining $\ell$-adic and Betti realizations. The source states that it follows from the first Tate conjecture together with injectivity of the comparison map, but provides no resolution status.