Single-polygon fully periodic symplectic billiards with unbounded periods

Determine whether there exists a single polygon P in the plane (allowing non-convexity) whose symplectic billiard map is fully periodic while admitting no uniform upper bound on the periods of its periodic orbits; equivalently, establish whether for every N there exists a periodic orbit of the symplectic billiard on P with period exceeding N.

Background

The paper introduces symplectic billiards for pairs of polygons and constructs many examples where every orbit is periodic. Among these, the authors provide the first example of a pair of polygons for which all orbits are periodic but the periods are unbounded, i.e., there is no uniform upper bound on the period.

In the classical single-polygon setting, several polygons (e.g., the Quad) are known to be fully periodic with a uniform period bound. However, no example is known where full periodicity holds without a uniform period bound, and it is explicitly stated that whether this phenomenon is possible for a single polygon remains unresolved.