The crossing conjecture for longest Hamiltonian cycles

Let $S$ be a point set in the plane in general position, and let a longest Hamiltonian cycle be a Hamiltonian cycle on $S$ whose total edge length is maximal. Longest-cycle crossing conjecture. Every longest Hamiltonian cycle of every point set has two edges that cross. The paper proves that a longest perfect matching can be chosen without crossings, but the corresponding question for Hamiltonian cycles remains open.