The finiteness conjecture for maximal totally geodesic submanifolds

Let $(M,g)$ be a closed Riemannian manifold with negative sectional curvature, of dimension $n\geq 3$. An immersed totally geodesic submanifold $N$ in $M$ is maximal if it is not contained in another proper closed immersed totally geodesic submanifold. The finiteness conjecture. If $M$ contains infinitely many maximal closed totally geodesic immersed submanifolds of dimension at least $2$, then $M$ is a locally symmetric space of rank $1$. This would generalize the main theorem and, together with known arithmeticity results, would imply that $M$ is arithmetic. The conjecture does not specify metric regularity; the authors suggest it may hold for $C^2$ metrics.