Asymptotic product conjecture for rainbow-triangle-free graph triples

Let $\gamma$ be the maximum on $[0,1]$ of

$$\frac{x^2}{2}\left(\frac{x^2}{2}+\frac{(1-x)^2}{2}\right)\left(x(1-x)+\frac{(1-x)^2}{2}\right),$$

and suppose that the maximum is attained at $x=x_0$. Consider three graphs $G_1,G_2,G_3$ on a common vertex set of size $n$, with no rainbow triangle. The asymptotic product conjecture. One has

$$e(G_1)e(G_2)e(G_3)\leq \gamma n^6(1+o(1)).$$

The paper proves a construction with product at least $\gamma n^6(1-o(1))$, showing that this is the natural asymptotic bound. Establishing the matching upper bound remains open in the source.