The entropy-or-flatness conjecture for compact manifolds without conjugate points

Let $M$ be a compact Riemannian manifold with no conjugate points, and let its geodesic flow be the natural flow on the unit tangent bundle of $M$.

Entropy-or-flatness conjecture. Either $M$ is flat or its geodesic flow has positive topological entropy.

This conjecture concerns classical Riemannian geometry and is motivated by questions about security. It predicts that, among compact manifolds without conjugate points, vanishing dynamical complexity occurs only in the flat case; the source presents it as an open conjecture.