Colliot-Thélène's exactness conjecture for zero-cycles

Let $k$ be a global field and let $X$ be a smooth, projective, geometrically integral variety over $k$. Write $CH_0(X)$ for the Chow group of zero-cycles modulo rational equivalence, and let $CH_0(X)^*$ denote the degree-zero-compatible profinite completion appearing in the source. Exactness conjecture for zero-cycles. The complex

$$\varprojlim_n CH_0(X)/n \longrightarrow \prod_v \varprojlim_n CH_0(X_{k_v})^*/n \longrightarrow {\rm Hom}({\rm Br}(X),\mathbb Q/\mathbb Z)$$

induced by the sum of the pairings of ${\rm Br}(X)$ with the groups $CH_0(X_{k_v})$, with values in ${\rm Br}(k_v)\subset\mathbb Q/\mathbb Z$, is exact. This conjecture encompasses the preceding conjecture on zero-cycles of degree one and concerns all smooth projective varieties without geometric restrictions. Its general status is open.