Locally conjugate Galois sections

Let X be a hyperbolic curve over a number field K and consider the short exact etale homotopy sequence associated to X. When v is a nonarchimedean valuation of K, we say that two splittings of this sequence are locally conjugate at v if their restrictions to a decomposition group of v are conjugate.

We are interested in the following problem: suppose that two splittings are locally conjugate for a `large' set of valuations, can we then deduce that they are globally conjugate?

In this talk we will discuss an approach to this problem which shows that with some additional assumptions the answer to the above question is positive. (this is a work in progress)