Local divisibility conjecture for Albanese kernels over number fields

Let $X$ be a smooth projective geometrically connected variety over a number field $K$ with a $K$-rational point. For a finite place $v$ of $K$, let

$$X_v:=X\times_K K_v$$

be the base change to the completion $K_v$, and suppose that the Albanese kernel decomposes as

$$T(X_v)=D_v\oplus F_v,$$

where $D_v$ is divisible and $F_v$ is finite. The local divisibility conjecture. The group $T(X_v)$ is divisible for almost all finite places $v$ of $K$, equivalently $F_v=0$ for almost all such places.

This is proposed as a weaker compatibility between the local and global pictures than the Bloch--Beilinson expectation that the global Albanese kernel is finite. The supplied text does not indicate whether the conjecture has been resolved.