The valuative section conjecture for curves

Let $k$ be a number field or a finite extension of $\mathbb Q_p$, and let $X/k$ be a smooth, projective and geometrically connected curve of genus at least $2$, with function field $K=k(X)$. Let $\widetilde K=k(\widetilde X)$ be the function field of the universal pro-étale cover, with Galois group identified with $\pi_1(X)$. For a valuation $w$ of $K$ and a prolongation $\widetilde w$ to $\widetilde K$, write $\operatorname{D}_{\widetilde w}$ for its decomposition subgroup. The valuative section conjecture. For every section $s:\operatorname{Gal}_k\to\pi_1(X)$ of $\pi_1(X/k)$, there exists $w\in\operatorname{Val}_k(K)$ with residue field $\kappa(w)=k$ and a prolongation $\widetilde w$ such that

$$s(\operatorname{Gal}_k)\subseteq \operatorname{D}_{\widetilde w}.$$

This valuative formulation asks whether every section is supported on a $k$-rational valuation. The paper proves a related valuative result for a broader valuation space, but the stated $\operatorname{Val}_k(K)$ conjecture is not established in general.