Tate's conjecture on algebraic cycles and Galois-fixed étale classes

Let $X$ be a smooth projective variety over a field $k$, let $l$ be invertible in $k$, and write $\bar{k}$ for an algebraic closure of $k$. Let $\rho^i(X\times_k\operatorname{Spec}\bar{k})_{\operatorname{\mathbb{Q}}_l}$ be the $l$-adic cycle class map and let $\operatorname{Gal}(\bar{k}/k)$ act on $\operatorname{H}^{2i}_{\operatorname{\text{\'e}t}}(X\times_k\operatorname{Spec}\bar{k},\operatorname{\mathbb{Q}}_l(i))$. Tate conjecture. For any $i$, the image of the cycle class map in this cohomology group is exactly the union of the fixed parts of open subgroups of $\operatorname{Gal}(\bar{k}/k)$. This is the $l$-adic formulation of the Tate conjecture and is open in general.