The endpoint monotonicity conjecture for polygonal lines of combinatorial Petrie matrices

Let $A\in \mathcal{CP}(n)$ and let $L$ be its corresponding polygonal line. If $n$ is even, consider the endpoints of $L$ ordered by increasing $x$-coordinate. Endpoint monotonicity conjecture. Their $y$-coordinates are weakly increasing in that order. The polygonal line is associated with a combinatorial Petrie matrix, and the claim concerns an additional monotonicity property beyond the established non-self-intersection and closed-curve properties. Its resolution is not indicated in the source.