Higher hyperbolic rank rigidity conjecture

A closed Riemannian manifold $M$ has higher hyperbolic rank if every geodesic $c(t)$ in $M$ has a Jacobi field $J(t)$ such that $J(t)\neq 0$ and the sectional curvature of the plane spanned by $J(t)$ and $c'(t)$ is $-1$. Assume that all sectional curvatures satisfy $\kappa\geq -1$. Higher hyperbolic rank rigidity conjecture. If $M$ has higher hyperbolic rank, then $M$ is locally symmetric. This conjecture proposes a converse to the fact that rank-one locally symmetric spaces of negative curvature have higher hyperbolic rank after rescaling the metric; it remains open in the stated generality.