The quadratic upper-bound conjecture for intersecting triangles

Let $S$ be a point set with $3n$ elements in the plane in general position. An intersecting edge-disjoint triangle family is a family of edge-disjoint triangles such that every two triangles have intersecting edges. Intersecting-triangle upper-bound conjecture. Every such point set has at most $n^2$ intersecting edge-disjoint triangles. The paper proves that $n^2$ triangles can be attained for point sets in convex position, so the conjecture asserts that this construction is extremal; it remains open in general.