Harborth's empty-triangle conjecture for simple drawings

For a simple drawing of $K_n$, let $\triangle(n)$ denote the minimum number of empty triangles among all such drawings. An empty triangle is a triangle induced by three vertices such that the interior of one of its two sides contains no vertex. Harborth's conjecture. It holds that $\triangle(n)=2n-4$. Harborth proved the bounds $2\leq\triangle(n)\leq 2n-4$; the optimality of the upper bound remains open.