Weak approximation for rationally connected varieties

Let $k$ be an algebraically closed field of characteristic zero, let $B$ be a curve over $k$, and let $k(B)$ be its function field. Let $X$ be a rationally connected variety over $k(B)$. Weak approximation conjecture. The variety $X$ satisfies weak approximation. The conjecture is motivated by the Kollár–Miyaoka–Mori theorem and existing weak approximation results for families with strongly rationally connected fibers; its general validity is left open here.