Periodicity conjecture for special octagons

A special octagon is a lattice octagon whose opposite sides are parallel with slopes $0$, $\pm1$, and $\infty$, and which has an axis of symmetry parallel to a pair of sides. Special-octagon periodicity conjecture. Every orbit of the symplectic billiard map in a special octagon is periodic. The paper gives two computational examples as evidence and describes the claim as a conjecture; its resolution is not supplied.