Rigidity of compactly supported perturbations of rank-one symmetric spaces

Let $(X,g_0)$ be a globally symmetric space of rank one with minimal curvature $-1$, and let $D$ be a compact subset. A Riemannian metric $g$ coincides with $g_0$ outside of $D$ if its restriction to $X\setminus D$ agrees with that of $g_0$.

Rigidity conjecture. Any Riemannian metric $g$ which coincides with $g_0$ outside of $D$ is isometric to $g_0$.

This asks whether rank-one symmetric spaces with minimal curvature $-1$ are rigid under compactly supported metric perturbations. The paper presents it as a question extending the rigidity results proved under additional curvature assumptions; its resolution is not supplied here.