Projective Blaschke conjecture

Let $M$ be a manifold with a normal projective connection, and suppose that its geodesics are embedded curves. Projective Blaschke conjecture. If all geodesics are closed embedded curves, then the projective connection is the standard normal projective connection of a compact rank one symmetric space. This is a projective analogue of the Blaschke problem; the supplied text does not state a resolution.