Tate-Beilinson-Parshin cycle map conjecture

Let $k$ be a finite field, let $\overline{k}$ be a separable closure, and let $G_k$ be its Galois group. Let $X$ be a projective smooth variety over $k$, let $\ell$ be a prime different from the characteristic of $k$, and let $\operatorname{CH}^i(X,j)$ denote Bloch's higher Chow group. Set $\overline{X}=X\times_k\overline{k}$. Tate-Beilinson-Parshin cycle map conjecture. For all integers $i,j\geq0$, the cycle map

$$\operatorname{CH}^i(X,j)\otimes\mathbb{Q}_{\ell}\longrightarrow H^{2i-j}(\overline{X},\mathbb{Q}_{\ell}(i))^{G_k}$$

is bijective. The paper explains that this combines the Tate-Beilinson and Parshin conjectures and holds for varieties in the class of projective smooth varieties of Abelian type for which the Tate conjecture holds; it remains open in general.