Single-polygon symplectic billiards with no periodic orbits

Determine whether there exists a single polygon P in the plane (allowing non-convexity) whose symplectic billiard map has no periodic orbits at all.

Background

The authors construct the ‘necktie’, the first pair of convex polygons whose symplectic billiard map has no periodic orbits. They contrast this with the smooth strictly convex two-curve setting, where infinitely many periodic orbits always exist.

Despite the pair-of-polygons example with no periodic orbits, it remains unknown whether a single polygon (the single-table case, P_- = P_+) can exhibit this phenomenon, and the authors explicitly identify this as an open question.