Dominated splitting conjecture for robustly transitive geodesic flows

Let $g$ be a Riemannian metric whose geodesic flow is robustly transitive, meaning that $g$ admits a $C^2$-neighbourhood such that every metric in that neighbourhood has topologically transitive geodesic flow. Dominated splitting conjecture. The geodesic flow of $g$ must admit a dominated splitting. This conjecture proposes an analogue of the characterization of robustly transitive dynamical systems in the setting of geodesic flows; the required perturbation techniques are expected to be fundamentally different. The source does not state whether it is resolved.