Termination of the intersecting-triangle matching algorithm

Prove that the iterative matching algorithm for a point set with half its points on the convex hull, which swaps the assignments of two points whenever the corresponding triangles over two convex-hull edges intersect, always terminates; equivalently, prove that the directed state graph of permutations defined by these swaps is acyclic for every such point set in general position.

Background

The proof that the double circle is universal requires finding a nonintersecting matching between interior points and convex-hull edges. The paper proposes a local improvement heuristic: whenever two associated triangles intersect, swap the two matched interior points.

Unlike the analogous red-blue point matching problem, the paper does not provide a monotone potential function for this point-to-edge variant. The authors verify the conjectured termination computationally for order types with up to ten points, but leave the general case unresolved.