The Brauer–Manin conjecture for zero-cycles of degree one

Let $k$ be a number field, and let $\{X_\omega\}_\omega$ be the family of all smooth, proper, geometrically integral varieties over $k$. The Brauer–Manin obstruction to weak approximation for 0-cycles of degree $1$ is defined using the Brauer–Manin pairing on adelic 0-cycles of degree $1$. Colliot-Thélène's conjecture. The Brauer–Manin obstruction to weak approximation for 0-cycles of degree $1$ is the only one for $\{X_\omega\}_\omega$. This is a central conjecture on the arithmetic of zero-cycles; it extends the analogous question for rational points to all smooth, proper, geometrically integral varieties and remains open in this generality.