Simple-trajectory conjecture for denominators of two-move riders

Let $\mathbf z$ be a vertex of an inside-out polytope for $q$ two-move riders, with components $z_i\in\mathbb R^2$. A simple trajectory is a primitive trajectory with no equal consecutive slopes and no segment on the board boundary. Simple-trajectory conjecture. Every component $z_i$ is determined by one of the following: a point along a $k$-point simple trajectory with $k\leq q$; an intersection of two simple trajectories having $i$ and $j$ points with $i+j\leq q+1$; or a self-intersection point of a $k$-point simple trajectory with $k\leq q$. Consequently, all denominators of $\mathbf z$ arise from such points. This is proposed as a description of the full scope of denominators for two-move riders, and remains open.