Harborth–Mengersen's uncrossed-edge conjecture for crossing-maximal drawings

A simple drawing of $K_n$ is crossing-maximal if it has $\binom{n}{4}$ crossings; an uncrossed edge is an edge crossed by no other edge. Harborth–Mengersen's conjecture. Every crossing-maximal simple drawing of $K_n$ contains an uncrossed edge. The conjecture is verified for $n\leq 16$ in the paper, but remains open in general.