On the 'Section Conjecture' in anabelian geometry

Let X be a smooth projective curve of genus > 1 over a field K with function field K(X), let pi(1)X be the arithmetic fundamental group of X over K and let G(F) denote the absolute Galois group of a field F. The section conjecture in Grothendieck's anabelian geometry says that the sections of the canonical projection pi(1)(X) -> G(K) are ( up to conjugation) in one-to-one correspondence with the K-rational points of X, if K is finitely generated over Q. The birational variant conjectures a similar correspondence w.r.t. the sections of the projection G(K(X)) -> G(K).