Quadratic-size $K_3$-intersecting family conjecture

Let $K_n$ be a complete geometric graph on $n$ points in general position in the plane. A family of subgraphs is $K_3$-intersecting if every two members intersect in a triangle. Quadratic-size $K_3$-intersecting family conjecture. $K_n$ contains a $K_3$-intersecting family of quadratic size, that is, a family whose size is $\Omega(n^2)$. This is proposed alongside the corresponding $P_3$ statement, following known lower bounds for other intersecting subgraphs; the conjecture is presented as open in the source.