Full-measure conjecture for closed geodesics on real affine structures

Let $\Sigma$ be a closed surface, and consider the finite-dimensional parameter space of real affine structures on $\Sigma$. A real affine structure carries a closed geodesic if it admits a closed geodesic. Full-measure conjecture. The set of real affine structures on $\Sigma$ carrying a closed geodesic has full measure in the parameter space of real affine structures on $\Sigma$. This conjecture concerns the prevalence of closed geodesics among real affine structures. The paper notes that little is known for general affine structures and that related questions, such as periodic billiard trajectories for arbitrary polygonal tables, are notoriously difficult.