Optimality conjecture for seven points spanning four distinct triangles

A finite point set in the plane spans four distinct triangles when its triples of distinct points determine four congruence classes, up to rigid plane isometries. Four-triangle optimality conjecture. The maximum number of points in the plane spanning four distinct triangles is $7$, and equality is achieved only by the regular heptagon. This is a natural next case after the solved three-triangle case; the source gives no resolution.