The arbitrarily long periodic-orbit conjecture for triangle tilings

A triangle tiling is a tiling of the plane by congruent triangles. A periodic trajectory is one that returns to its initial position and direction, and its length is the number of reflections or segments in one period.

Triangle periodic-length conjecture. There exist triangle tilings with periodic trajectories of arbitrarily large length.

The paper gives examples of periodic trajectories, including a trajectory of period $34$, but the existence of arbitrarily large periods remains open.