The Brauer–Manin zero-cycle conjecture for smooth projective varieties

Let $X$ be a smooth, projective, geometrically integral variety over a number field $k$. Let

$$X({\bf A}_k)^{\operatorname{Br}}$$

denote the Brauer–Manin set of adelic points orthogonal to $\operatorname{Br}(X)$, and let $I(X)$ be the index of $X$, namely the greatest common divisor of the degrees of closed points of $X$.

Brauer–Manin zero-cycle conjecture. If $X({\bf A}_k)^{\operatorname{Br}}$ is non-empty, then $I(X)=1$.

The analogous assertion for rational points is false for arbitrary smooth projective varieties, but this weaker zero-cycle statement remains open in general.