The divisible-plus-finite conjecture for the Albanese kernel of a smooth projective variety

Let $X$ be a smooth projective variety over a finite extension of the $p$-adic field $\mathbb{Q}_p$. Let $F^2(X)$ denote the kernel of the Albanese map on the Chow group of zero-cycles.

Divisible-plus-finite conjecture. The group $F^2(X)$ is the direct sum of a divisible group and a finite group.

This conjecture predicts the structure of the Albanese kernel over $p$-adic fields and is attributed in the source to work of Raskind–Spiess and Colliot-Thélène. The supplied text gives no evidence that it has been resolved.