Geodesic rigidity conjecture for projectively equivalent metrics

Let $M^n$ be a closed connected manifold, and let $g$ and $\bar g$ be Riemannian metrics on $M^n$ that are projectively equivalent and nonproportional. Geodesic rigidity conjecture. The manifold $M^n$ can be covered by the sphere, or it admits a local product structure. The conjecture concerns the global classification of manifolds carrying nonproportional projectively equivalent metrics; the surrounding discussion identifies this as an unresolved global question, while noting that products of spheres provide examples and that a related finite-cover theorem is proved for metrics strictly non-proportional at at least one point.