Let $X$ be a finite dimensional Alexandrov space with curvature bounded below and non-empty boundary. Let $\gamma$ be a shortest geodesic path in $X$ whose endpoints belong to the interior of $X$.

Is it true that $\gamma$ is contained in the interior of $X$? Is that true at least under the assumption that $X$ is smooth Riemannian manifold with smooth boundary (thus $X$ must be locally geodesically convex)?