Beilinson's Albanese isomorphism conjecture for zero-cycles

Let $K$ be a global field with separable closure $K^s$, and let $X^s$ be a smooth projective variety over $K^s$. Let $\operatorname{CH}_0(X^s)^0$ denote the degree-zero Chow group of zero-cycles and let $\operatorname{Alb}_{X^s}$ be the Albanese variety. Beilinson's conjecture. The Albanese map

$$\operatorname{CH}_0(X^s)^0\longrightarrow \operatorname{Alb}_{X^s}(K^s)$$

is an isomorphism. This conjecture is used to prove finiteness of the cokernel of $\rho_d$; the source states that Beilinson formulated it for number fields, while here it is considered over global fields.