Nonlinear-escape conjecture for bounded locally foldable tilings

Fix the combinatorics of a periodic $2$-colored tiling of a plane, defined by a bipartite graph $G$ on the $2$-torus. Suppose that a tiling with these combinatorics folds into a bounded domain. Nonlinear-escape conjecture. Nonlinear escape of trajectories on such a tiling is possible only when the trajectories pass through the point $C$; otherwise, trajectories either escape linearly or are periodic. The conjecture is motivated by the codimension-one theory and the helicoid construction, but the source notes that regularity issues remain because locally foldable tilings may have saddles of higher multiplicity than those allowed by the Morse assumptions used in the cited theory.