The conjecture that every divisible-order complete graph has a 3-symmetric optimal drawing

Let $K_n$ be the complete graph drawn geometrically, with vertices represented by points in general position and edges by straight-line segments. A drawing is optimal if it has the minimum possible number of crossings among geometric drawings of $K_n$, and it is 3-symmetric if it has 3-fold symmetry.

3-symmetric optimal-drawing conjecture. For each positive integer $n$ divisible by $3$, there is an optimal geometric drawing of $K_n$ that is 3-symmetric.

The conjecture is suggested by the resemblance between known optimal constructions and 3-symmetric point sets, although the paper notes that previously known explicit best constructions were generally not 3-symmetric. The assertion remains open.