Hassett–Tschinkel weak approximation conjecture for rationally connected varieties

Let $k$ be an algebraically closed field of characteristic zero, let $B$ be a smooth projective algebraic $k$-curve, and let $F$ be its function field. A proper rationally connected variety is a variety that is proper and rationally connected over $F$.

Hassett–Tschinkel's weak approximation conjecture. Proper rationally connected varieties defined over $F$ satisfy weak approximation.

This conjecture concerns the arithmetic of rationally connected varieties over function fields of curves. It was proposed by Hassett and Tschinkel and is presented here as motivation for studying analogous questions for separably $\mathbb{A}^1$-connected varieties; its resolution status is not specified in the source.