The flat-geodesic finiteness and ergodicity conjecture for nonpositively curved surfaces

Let $(M,g)$ be a smooth, connected and closed surface of genus $\mathfrak{g}\geq 2$ with nonpositive curvature. A flat geodesic is a geodesic whose curvature vanishes along its entire trajectory; let $\Lambda$ be the set of their unit tangent vectors, and let $\nu$ denote the natural invariant measure on $SM$.

Flat-geodesic finiteness conjecture. All flat geodesics are closed, and there are only finitely many homotopy classes of such geodesics. In particular,

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

and the geodesic flow on $SM$ is ergodic.

The conjecture addresses whether the singular or flat geodesics form a negligible set and whether every such geodesic is periodic. The supplied source gives no resolution, so its status remains open.