Colliot-Thélène's local-to-global conjecture for zero-cycles

For a number field $K$, let $X$ be a smooth projective geometrically connected variety defined over $K$. Let $\operatorname{CH}_0(X)$ be the Chow group of zero-cycles, let $\operatorname{CH}_{0,\mathbb A}(X)$ be its adelic analogue, and write $\widehat{A}=\varprojlim_n A/nA$ for the profinite completion of an abelian group $A$. The diagonal map induces a complex

$$\widehat{\operatorname{CH}_0(X)}\xrightarrow{\Delta}\widehat{\operatorname{CH}_{0,\mathbb A}(X)}\xrightarrow{\varepsilon}\operatorname{Hom}(\operatorname{Br}(X),\mathbb{Q}/\mathbb{Z}).$$

Colliot-Thélène's conjecture. The complex above is exact.

This conjecture asserts that the Brauer–Manin obstruction is the only obstruction to the local-to-global principle for zero-cycles. It is a central zero-cycle analogue of the corresponding principle for rational points; the supplied text does not state a resolution.