The security conjecture for closed Riemannian manifolds

Let $(M,g)$ be a closed Riemannian manifold. A pair of points is secure if there is a finite set, disjoint from the pair, meeting every geodesic segment joining them; $M$ is secure if every pair is secure. A Riemannian manifold is flat if its sectional curvature vanishes identically.

Security conjecture. A closed Riemannian manifold is secure if and only if it is flat.

Flat manifolds are uniformly secure, and the conjecture has been verified in several settings, including manifolds without conjugate points, compact locally symmetric spaces, and compact Riemannian surfaces of genus at least one. The general case remains open.