On the birational section conjecture with strong birationality assumptions

Let $X$ be a hyperbolic curve over a field $k$ finitely generated over $mathbb{Q}$. A Galois section $s$ of $pi_{1}(X)tomathrm{Gal}(bar{k}/k)$ is birational if it lifts to a section of $mathrm{Gal}(overline{k(X)}/k(X))tomathrm{Gal}(bar{k}/k)$. Grothendiecks section conjecture predicts that every Galois section of $pi_{1}(X)$ is either geometric or cuspidal, while the birational section conjecture predicts the same for birational Galois sections. Let $t$ be an indeterminate. We prove that, if $s$ is a Galois section such that the base change $s_{k(t)}$ to $k(t)$ is birational, then $s$ is geometric or cuspidal. As a consequence we prove that the section conjecture is equivalent to Esnault and Hais cuspidalization conjecture, which states that Galois sections of hyperbolic curves over fields finitely generated over $mathbb{Q}$ are birational.