The birational section conjecture over finitely generated fields

Let $k$ be a finitely generated field over $\operatorname{\mathbb{Q}}$, and let $X$ be a smooth, projective, geometrically connected curve over $k$. Write $G_X:=\operatorname{Gal}(\overline{k(X)}/k(X))$ for the absolute Galois group of $X$, and call a section of the exact sequence $1\to G_{X_{\bar k}}\to G_X\to G_k\to 1$ geometric if its image is contained in a decomposition group associated with a $k$-rational point of $X$. Birational section conjecture. Every section of $G_X$ is geometric and arises from a unique $k$-rational point $x\in X(k)$. The paper investigates this conjecture over function fields of characteristic zero and proves that it holds over finitely generated fields over $\operatorname{\mathbb{Q}}$ if it holds over number fields; the general assertion remains open.