Locally conjugate Galois sections

We consider sections of the étale homotopy exact sequence of a hyperbolic curve over a number field. We prove that two sections whose restrictions to decomposition groups are conjugate on a set of valuations of density one are globally conjugate, which establishes the local-global principle for the conjugacy classes of sections. In fact, we obtain this result as a corollary of a more general property concerning sections of the étale homotopy exact sequence, the finite covering property, which we prove as our main result.