Decreasing intersection quotients characterize quasi-geodesic origami edge-paths

Let $\alpha$ and $\beta$ be the endpoints of an origami edge-path

$$\{\alpha=\alpha_0,\ldots,\alpha_{n>2}=\beta\}.$$

Assume that the successive intersection quotients satisfy

$$\frac{|\beta\cap\alpha_1|}{|\alpha\cap\alpha_1|}>\frac{|\beta\cap\alpha_2|}{|\alpha\cap\alpha_2|}>\cdots>\frac{|\beta\cap\alpha_n|}{|\alpha\cap\alpha_n|}.$$

Intersection-quotient conjecture. Under this condition, the edge-path is a quasi-geodesic.

The quotients can be viewed as slopes in the flat structure determined by the $(\alpha,\beta)$-origami. Strict decrease is intended to prevent zig-zagging near a geodesic. The source gives no resolution of the conjecture.