Totally geodesic rigidity conjecture for negatively curved manifolds

Let $M$ be a compact Riemannian manifold with negative sectional curvature and dimension at least $3$. Suppose that, for some $k\geq 2$, there exist infinitely many distinct immersed maximal totally geodesic $k$-dimensional submanifolds of $M$. Totally geodesic rigidity conjecture. Then $M$ is isometric to a negatively curved locally symmetric space, namely $\Lambda\backslash G/K$, where $G$ is a real rank-$1$ Lie group, $K\subset G$ is a maximal compact subgroup, and $\Lambda$ is a lattice in $G$. This is posed as evidence for a broader rigidity principle relating infinitely many totally geodesic submanifolds to homogeneous geometry; the supplied text does not state a resolution.