A closed contact cycle on the ideal trefoil

Numerical computations suggest that each point on a certain optimized shape called the ideal trefoil is in contact with two other points. We consider sequences of such contact points, such that each point is in contact with its predecessor and call it a billiard. Our numerics suggest that a particular billiard on the ideal trefoil closes to a periodic cycle after nine steps. This cycle also seems to be an attractor: all billiards converge to it.