Beilinson's Tate conjecture for regulator maps

Let $U$ be a smooth variety over a field $k$ of characteristic zero. When $k$ is finitely generated over $\mathbb Q$, there is an étale regulator map

$$\operatorname{reg}_{U,\mathrm{et}}^q: CH^q(U,q)\otimes\mathbb Q_{\ell}\to H^q_{\mathrm{et}}(U_{\overline{k}},\mathbb Q_{\ell}(j))^{\operatorname{Gal}(\overline{k}/k)}.$$

Here $CH^q(U,q)$ is the higher Chow group, $U_{\overline{k}}=U\times_k\overline{k}$, and $\overline{k}$ is an algebraic closure of $k$.

Beilinson's Tate conjecture. In case $k$ is finitely generated over $\mathbb Q$, $\operatorname{reg}_{U,\mathrm{et}}^q$ is surjective.

This is an analogue for open varieties of the Tate conjecture for algebraic cycles on smooth projective varieties. The conjecture holds for $q=1$, while the general case remains open.