The boundary-point conjecture for the Heilbronn triangle problem

Consider the Heilbronn triangle problem for $n$ points in the unit square $[0,1]^2$. An optimal solution is a point configuration attaining the maximum possible minimum area of a triangle determined by three points. The boundary of the square consists of its four edges.

Boundary-point conjecture. For any optimal solution with at least nine points, each edge of the unit square $[0,1]^2$ contains exactly two points.

Computational experiments and all best-known configurations reported in the cited literature exhibit eight boundary points for configurations with at least nine points, motivating this conjecture. The corresponding property is proved for seven and eight points, while a mathematical proof for nine points is not known.