Chow-group identification conjecture for smooth proper varieties

Let $k\subset F$ be the fields in the paper, let $n=\infty$, and let $X$ be a smooth irreducible proper variety over $k$. Write $CH_0(X_F)_{\mathbb Q}$ for the rationalized group of zero-cycles on $X_F$, and let $C_{k(X)}$ denote the associated object defined from the function field $k(X)$. Chow-group identification conjecture.

$$CH_0(X_F)_{\mathbb Q}=C_{k(X)}.$$

This would identify the rational zero-cycle group after base change with the function-field construction used in the paper; the source provides no resolution of the claim.