Tree conjecture for cyclic quadrilateral tiling billiards

Let $\gamma$ be a periodic trajectory in a cyclic quadrilateral tiling, and let $\Omega_{\gamma}$ be the domain bounded by it. Tree conjecture for cyclic quadrilateral tiling billiards. Any periodic trajectory $\gamma$ in a cyclic quadrilateral tiling does not contour tiles; equivalently, the domain $\Omega_{\gamma}$ bounded by it does not contain a full tile. This conjecture concerns the symbolic dynamics of cyclic quadrilateral tilings and is related, for locally foldable tilings, to the bounded-flower formulation involving singular trajectories.