Crossing-maximal large crossing-family conjecture

A $k$-crossing family is a set of $k$ independent edges that cross pairwise. A simple drawing of $K_n$ is crossing-maximal when it has the maximum possible number $\binom{n}{4}$ of crossings. Large crossing-family conjecture. Every crossing-maximal drawing of $K_n$ contains a $\left\lfloor\frac{n}{2}\right\rfloor$-crossing family. The statement is presented as an expected property of crossing-maximal drawings, with no resolution given in the paper.