Beilinson–Bloch conjecture on the Albanese kernel

Let $K$ be a number field and let $Z/K$ be a smooth projective irreducible variety of dimension $d$. Define the Albanese kernel by

$$T(Z):=\operatorname{Ker}\left(\mathrm{CH}^d(Z)_0\to\operatorname{Alb}(Z)\right).$$

Here $\mathrm{CH}^d(Z)_0$ is the group of homologically trivial zero-cycles and $\operatorname{Alb}(Z)$ is the Albanese variety. Beilinson–Bloch conjecture. The Albanese kernel $T(Z)$ is a finite group. This is a special case of the Beilinson–Bloch conjectures concerning zero-cycles on varieties over number fields; the supplied text does not state whether this case has been resolved.