Quadratic lower-bound conjecture for empty red-red-blue triangles

Let $R$ and $B$ be disjoint sets of $n$ red and $n$ blue points, respectively, in the plane. An empty red-red-blue triangle is a triangle whose vertices consist of two points of $R$ and one point of $B$, and whose interior contains no point of $R\cup B$. Quadratic lower-bound conjecture. There exist $\Omega(n^2)$ empty red-red-blue triangles in $R\cup B$. The paper notes that no construction with subquadratically many such triangles is known; in particular, balanced bichromatic Horton sets contain $\Omega(n^2)$ of them. Thus the conjecture remains open.