On the birational section conjecture with strong birationality assumptions

Let X be a curve over a field k finitely generated over ℚ and t an indeterminate. We prove that, if s is a section of π1(X) → Gal (k) such that the base change sk(t) is birationally liftable, then s comes from geometry. As a consequence we prove that the section conjecture is equivalent to the cuspidalization of all sections over all finitely generated fields.