Schoen's integral Tate conjecture

Let $X$ be a smooth projective variety over an algebraically closed field $k$, let $\ell\neq\operatorname{char}(k)$ be a prime, and choose a subfield $k_0\subseteq k$ over which $X$ is defined with $\overline{k_0}=k$. Write

$$\operatorname{cl}^i_{\mathbb{Z}_\ell}\colon \operatorname{CH}^i(X)_{\mathbb{Z}_\ell}\to\mathrm{H}_{\acute{e}t}^{2i}(X,\mathbb{Z}_\ell(i))$$

for the $\ell$-adic cycle class map. For finite extensions $k'$ of $k_0$, let $G_{k'}$ denote the absolute Galois group of $k'$. Schoen's integral Tate conjecture. If $k_0$ is finite over its prime field, then

$$\operatorname{im}(\operatorname{cl}^i_{\mathbb{Z}_\ell})=\varinjlim_{k_0\subseteq k'}\mathrm{H}_{\acute{e}t}^{2i}(X,\mathbb{Z}_\ell(i))^{G_{k'}}.$$

This is the integral Tate analogue of the Hodge conjecture over fields of positive characteristic. The source gives no resolution status for this assertion.