Bergold et al.'s empty-cycle conjecture for simple drawings

An empty $k$-cycle in a simple drawing of $K_n$ is a plane cycle through $k$ vertices such that all remaining vertices lie on one common side. Bergold et al.'s conjecture. Every simple drawing of $K_n$ contains an empty $k$-cycle for every $k=3,\ldots,n$. The conjecture is proved for convex drawings and verified for simple drawings with $n\leq 10$; the general case remains open.