Removing central symmetry in the Bialy–Mironov rigidity theorem

Determine whether Theorem \ref{thm:birkhoff-1/4} (rigidity from total integrability on a boundary-adjacent region along with an invariant curve of 4-periodic orbits) holds without the assumption of central symmetry of the billiard table.

Background

Theorem \ref{thm:birkhoff-1/4} shows that, for centrally symmetric tables, the combination of a 4-periodic invariant curve and either total integrability on a specific region or local maximality of orbits forces the boundary to be an ellipse.

The authors ask whether this central symmetry assumption can be eliminated while retaining the conclusion.