Bergold et al.'s plane subdrawing conjecture for simple drawings

Let $K_n$ be drawn simply in the plane. A plane subdrawing is a subdrawing whose edges do not cross, and a Hamiltonian cycle is a cycle through all $n$ vertices. Bergold et al.'s conjecture. Every simple drawing of $K_n$ contains a plane subdrawing with $2n-3$ edges which contains a Hamiltonian cycle. The conjecture is proved for convex drawings and verified for simple drawings with $n\leq 9$; it remains open in general.