Extension of the hyperbolic closed-geodesic characterization to reversible metrics

Let $M$ be a closed manifold. A Finsler metric on $M$ is reversible if its associated norm is invariant under reversing tangent vectors, and a Riemannian metric is understood in the usual sense. For either class of metrics, consider the metrics all of whose closed geodesics are hyperbolic, and their $C^2$ interior.

Hyperbolic closed-geodesic characterization conjecture. The characterization theorem for Finsler metrics, and for Riemannian metrics on closed surfaces, should also hold for reversible Finsler metrics or Riemannian metrics on closed manifolds of arbitrary dimension: the $C^2$ interior of the metrics all of whose closed geodesics are hyperbolic should be exactly the set of metrics whose geodesic flow is Anosov.

The theorem established in the paper gives this characterization for general Finsler metrics in the stated setting and for Riemannian metrics on closed surfaces. The conjecture proposes the corresponding result for reversible Finsler and Riemannian metrics on closed manifolds in arbitrary dimension.