Bergold–Fabila-Monroy–Roldán-Pensado–Schaefer conjecture on empty cycles in simple drawings

A simple drawing of $K_n$ is a drawing of the complete graph in the plane with no self-intersecting edges, no crossings between adjacent edges, and at most one crossing between any pair of nonadjacent edges. A plane cycle is a cycle whose edges do not cross, and it divides the plane into two sides. An empty $k$-cycle is a plane cycle of length $k$ such that one of its sides contains no vertices in its interior. Bergold, Fabila-Monroy, Roldán-Pensado, and Schaefer conjectured that

$$\text{every simple drawing of }K_n\text{ contains an empty }k\text{-cycle for each }3\leq k\leq n.$$

Bergold–Fabila-Monroy–Roldán-Pensado–Schaefer conjecture. Every simple drawing of $K_n$ contains an empty $k$-cycle for each $3 \leq k \leq n$. The supplied paper proves the case $k=4$, even through every vertex, while the full assertion remains open there; the statement is known for convex drawings.