Ergodicity conjecture for geodesic flows on surfaces with nonpositive curvature

Let $(M,g)$ be a smooth, connected and closed surface of genus $d524\geq 2$ with nonpositive curvature. Let $SM$ be its unit tangent bundle, let $\Lambda$ denote the set of flat geodesics, and let $\nu$ denote Liouville measure. Ergodicity conjecture. All flat geodesics are closed, there are only finitely many homotopy classes of such geodesics, and consequently

$$\nu(\Lambda)=0,$$

so that the geodesic flow on $SM$ is ergodic. The conjecture concerns the expected smallness of the flat-geodesic set and its implication for ergodicity. The source notes that some experts expect a negative answer, while the paper proves the stated conclusions under an additional assumption.