Beilinson's Abel-Jacobi and Galois-invariance conjecture

Let $r\geq 2$ be an integer. For a nonsingular projective variety $X$ over $\overline{\bold Q}$, let

$$\rho:{\mathrm{CH}}^r(X)_{\hom}\longrightarrow J^r(X_{\bold C})$$

be the Abel-Jacobi map. If $X_{\bold C}=X\otimes_{\overline{\bold Q}}\bold C$ and $z\in {\mathrm{CH}}^r(X_{\bold C})$, write $z^\sigma$ for its conjugate under $\sigma\in\operatorname{Aut}(\bold C/\overline{\bold Q})$. Beilinson's conjecture. (i) The map $\rho$ is injective for every such $X$. (ii) For every such $X$, $z$, and $\sigma$,

$$\rho(z)=0\quad\Longleftrightarrow\quad\rho(z^\sigma)=0.$$

The first assertion predicts injectivity of the Abel-Jacobi map on homologically trivial cycles, while the second predicts invariance of vanishing under field automorphisms. The source gives no resolution status.