Colliot-Thélène's approximation conjecture for zero-cycles

Let $X$ be a smooth, projective, geometrically integral variety of dimension $d$ over a number field $k$. For each place $v$ of $k$, write $X_v=X\times_k k_v$, let $Z_0(X_v)$ be the group of zero-cycles, and let $\Omega_k$ be the set of places of $k$. For an integer $\delta$, a family $\{z_v\}\in\prod_{v\in\Omega_k}Z_0(X_v)$ has degree $\delta$ and is orthogonal to the Brauer group $Br(X)$ when its Brauer–Manin pairing with every element of $Br(X)$ is zero. Let $\widetilde{H}^{2d}(X_v,\mathbb{Z}/m\mathbb{Z}(d))$ be the modified étale cohomology group appearing in the local cycle class map.

Colliot-Thélène's approximation conjecture. If $\{z_v\}\in\prod_{v\in\Omega_k}Z_0(X_v)$ is a degree-$\delta$ family orthogonal to $Br(X)$, then for every positive integer $m$ there exists a global zero-cycle $z_m\in Z_0(X)$ of degree $\delta$ such that, at every place $v$, the cycles $z_m$ and $z_v$ have the same image under

$$CH_0(X_v)\to\widetilde{H}^{2d}(X_v,\mathbb{Z}/m\mathbb{Z}(d)).$$

This is a stronger approximation statement than mere existence of a global zero-cycle: it requires simultaneous congruence in the local cycle-class groups for every modulus $m$. The supplied source attributes the conjectural statements to Colliot-Thélène; no resolution status is given here.