Let $X$ be a simply connected manifold with nonpositive sectional curvature. It is standard that $X$ is uniquely geodesic, i.e., for any distinct points $p$ and $q$, there is a unique geodesic connecting them.

Now let us take a compactification of $X$ such that $\partial X$ is homeomorphic to $S^{n - 1}$. I'm interested in the uniqueness property on $\overline{X} = X \cup \partial X$: for any distinct points $p,q\in \partial X$, is it true that there is a unique geodesic $\gamma \subset X$ connecting the two points at infinity?