Conjecture (E) for zero-cycles on rationally connected varieties

Let $X$ be a smooth proper variety over a number field $k$. Let $\Omega_f$ be the set of finite places of $k$ and $\Omega_\infty$ the set of infinite places. For each infinite place $v$, let $\overline{k}_v$ be an algebraic closure of $k_v$, and let $N_{\overline{k}_v/k_v}$ denote the norm map. Consider the complex

$$\mathrm{CH}_0(X) \longrightarrow \prod_{v \in \Omega_f} \mathrm{CH}_0(X_{k_v}) \times \prod_{v \in \Omega_\infty} \frac{\mathrm{CH}_0(X_{k_v})}{N_{\overline{k}_v/k_v}(\mathrm{CH}_0(X_{\overline{k}_v}))} \longrightarrow \operatorname{Hom}(\operatorname{Br}(X),{\mathbf Q}/{\mathbf Z}).$$

The second map is the sum of the local pairings of zero-cycles with Brauer classes, followed by the local class-field-theoretic invariant. Conjecture (E). If $X$ is rationally connected, this complex is exact.

This is the zero-cycle analogue of the rational-points conjecture above and is attributed to Colliot-Thélène, Sansuc, Kato and Saito. It expresses that the Brauer–Manin pairing gives the only obstruction to the expected local-to-global exactness for zero-cycles on rationally connected varieties; the general assertion remains open.