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 introduce the following problem: suppose that two splittings are locally conjugate for a "large" set of valuations, can we then deduce that they are locally conjugate for all valuations? In this talk we will discuss a few positive results in this direction.