The -adic Tate conjecture for divisors

Let $k$ be a finitely generated field, let $X$ be a smooth and proper $k$-variety, and let $\ell$ be a prime different from the characteristic of $k$. Write $X_{\overline{k}}=X\times_k\overline{k}$ and let $\pi_1(k)$ denote the absolute Galois group of $k$. The $\ell$-adic cycle class map is

$$c_{X_{\overline{k}}}:\operatorname{Pic}(X_{\overline{k}})\otimes \mathbb{Q}_{\ell}\longrightarrow \bigcup_{[k':k]<+\infty}H^2(X_{\overline{k}},\mathbb{Q}_{\ell}(1))^{\pi_1(k')}$$

The $\ell$-adic Tate conjecture for divisors. This map should be surjective. The conjecture predicts that divisor classes account for all cohomology classes fixed by the absolute Galois group after a finite extension of the ground field, and is used to study the finiteness of Galois-invariant prime-to-characteristic torsion in geometric Brauer groups.