The conjecture that optimal rectilinear drawings of $K_n$ are 3-decomposable

For a positive integer $n$, let $K_n$ be the complete graph drawn with straight-line edges on a point set in general position. A drawing is optimal if it minimizes the number of crossings among rectilinear drawings of $K_n$, and it is 3-decomposable if its underlying point set admits the three-part decomposition described in the paper.

3-decomposability conjecture. For each positive integer $n$ divisible by $3$, all optimal rectilinear drawings of $K_n$ are 3-decomposable.

The conjecture is motivated by the fact that all best crossing-wise known drawings have underlying point sets minimizing the number of $(\leq k)$-sets for every $k\leq n/3$, a property equivalent to 3-decomposability. It remains open in general.