To what extent are open subsets of a hyperbolic pair of pants determined by the metric?

Question

Let $P$ be a pair of pants (a sphere with three open discs removed) and let $g_1,g_2$ be two Riemannian metrics on $P$ of constant Gaussian curvature $K \equiv -1$ such that the three boundary components of $P$ are geodesics. If the identity map $P \to P$ restricts to an isometry $(U,g_1) \to (U,g_2)$ for some open subset $U$, is the identity map necessarily an isometry $(P,g_1) \to (P,g_2)$?