Shortest origami edge-paths minimize successive endpoint intersections

Let $\alpha$ and $\beta$ be the endpoints of a shortest origami edge-path, and let $\alpha_i$ and $\alpha_{i+1}$ be consecutive curves on it, with $\alpha_{i+1}$ intersecting $\alpha_i$ once. Intersection-minimization conjecture. The quantity $|\beta\cap\alpha_{i+1}|$ is minimal among all curves $\alpha_{i+1}$ intersecting $\alpha_i$ once.

This conjecture formalizes the intuition that shortest paths should move monotonically toward the endpoint $\beta$, rather than zig-zagging. The source gives no resolution of the claim.