Disjoint-triangles conjecture for decompositions of complete geometric graphs

Let $\mathsf{K}_n$ be a complete geometric graph and let $[\mathsf{K}_n,P]$ be a decomposition. Disjoint-triangles conjecture. If $P$ contains at least

$$\left(\frac{n}{3}\right)^2+2$$

triangles, then $P$ contains at least two disjoint triangles. The conjecture is motivated by extremal decompositions with large pairwise-intersecting families of triangles; its status is open.