Beilinson's conjecture for higher Chow groups

Let $X$ be a smooth quasi-projective variety over a finite field $\mathbb F$. The higher cycle map is

H^p_M(X,\mathbb Z(q))\otimes_{\mathbb Z}\mathbb Q_\ell\xymatrix@1@=15pt{\ar[r]&} H^p(\bar X,\mathbb Q_\ell(q))^{G_{\mathbb F}}.

Beilinson's conjecture for higher Chow groups. This map should be injective.

Together with the Tate conjecture for higher Chow groups, this predicts that the map is an isomorphism; the general statement remains open.