Frankl's product conjecture for rainbow-triangle-free graph triples

Let $G_1,G_2,G_3$ be graphs with a common vertex set on $n$ vertices. A rainbow triangle is a triangle whose three edges belong to three distinct graphs among $G_1,G_2,G_3$. Frankl's conjecture. If the graphs have no rainbow triangle, then

$$e(G_1)e(G_2)e(G_3)\leq \left\lfloor\frac{n^2}{4}\right\rfloor^3.$$

The bound is attained by taking three copies of a complete bipartite graph with parts as equal as possible. Frankl proved the conjecture under the additional assumptions $E(G_1)\subseteq E(G_2)$ and $E(G_1)\subseteq E(G_3)$, but the paper gives a counterexample in general, so the conjecture is false.