Hausdorff dimension and dense-orbit conjecture for polygonal outer length billiards

Let $Q$ be a polygon, and let its singularity set be the set of points where the outer length billiard is singular. Let an orbit be a sequence of points generated by the billiard, and let its closure be the closure of that sequence. Singularity-set conjecture. The singularity set of every polygon has Hausdorff dimension greater than $1$. Furthermore, for every polygon, there exists an orbit whose closure is the singularity set, and which is therefore unbounded. The conjecture is motivated by computer experiments showing apparently dense orbits near the singularity set; its general validity remains open.