Hoffmann–Tóth convex matching conjecture for simple drawings

Let $K_n$ have a convex drawing, and let $M$ be a plane matching, meaning a set of pairwise vertex-disjoint edges no two of which cross. Hoffmann–Tóth convex-matching conjecture. For every plane matching $M$ in a convex drawing of $K_n$, there exists a plane Hamiltonian cycle that does not cross any edge from $M$. The conjecture is verified for $n\leq 11$ by the paper's computations, but remains open beyond that range.