Measure-zero conjecture for nonlinear escape in locally foldable tilings

Fix the combinatorics $G$ of a periodic $2$-colored tiling of the plane, and consider the parameters of locally foldable tilings with those fixed combinatorics. Measure-zero conjecture for nonlinear escape. The set of parameters admitting nonlinearly escaping trajectories has measure zero. This is presented as a stronger reformulation of an earlier question; the preceding nonlinear-escape conjecture and the required regularity arguments are not established in the source.