Colliot-Thélène's conjecture on rationally connected varieties

Let $k$ be a number field, let $\Omega$ be the set of places of $k$, and write $k_{\Omega}=\prod_{v\in\Omega}k_v$. For a smooth $k$-variety $X$, let $\operatorname{Br}_{\mathrm{nr}}(X)$ denote its unramified Brauer group and let $X(k_{\Omega})^{\operatorname{Br}_{\mathrm{nr}}}$ be the left kernel of the Brauer–Manin pairing.

Colliot-Thélène's conjecture. Let $X$ be a rationally connected smooth variety over $k$. Then $X(k)$ is dense in $X(k_{\Omega})^{\operatorname{Br}_{\mathrm{nr}}}$.

This conjecture predicts that rational connectedness, together with the absence of a Brauer–Manin obstruction, controls the existence and approximation of rational points. It would imply important arithmetic consequences, including a positive answer to the inverse Galois problem via versal torsors. Its status is not resolved in the supplied source.