Local symmetry conjecture for finite-volume manifolds satisfying (P)

Let $(N,g)$ be a finite-volume Riemannian manifold satisfying property (P), meaning that every geodesic in its universal covering has rank at least two. Local symmetry conjecture. The manifold $(N,g)$ is locally symmetric. This is motivated by hyperbolic rank-rigidity results for finite-volume manifolds; the conjecture asks whether the corresponding rigidity persists under property (P) in this setting.