Frankl–Győri–He–Lv–Salia–Tompkins–Varga–Zhu conjecture for Gallai 3-colouring templates

Let $h:[0,1]\to\mathbb{R}$ be defined by

$$h(x)=\left(x^2+(1-x)^2\right)x^2\left(1-x^2\right),$$

and let $\upsilon\in[0,1]$ maximise $h$. Let $\mathbf{G}=(G_1,G_2,G_3)$ be a Gallai $3$-colouring template on $n$ vertices, meaning that its three colour classes partition the pairs of vertices and contain no rainbow triangle.

Frankl–Győri–He–Lv–Salia–Tompkins–Varga–Zhu conjecture. As $n$ tends to infinity,

$$\Bigl(\lvert E(G_1)\rvert\cdot\lvert E(G_2)\rvert\cdot\lvert E(G_3)\rvert\Bigr)^{\frac13}\leq\left(h(\upsilon)+o(1)\right)^{\frac13}\binom{n}{2}.$$

The conjecture asserts that the construction $\mathbf{H}(\lceil\upsilon n\rceil,n-\lceil\upsilon n\rceil,0)$ is asymptotically optimal. It was proposed after that construction disproved Frankl's earlier upper bound; the supplied text gives no resolution of this conjecture.