The Strong Erdős–Szemerédi conjecture for sums and products along graph edges

Let $G_n$ be a graph on $n$ vertices, and let $\mathcal{A}\subset\mathbb{R}$ be an $n$-element set. Define the sumset and product set along $G_n$ by

$$\mathcal{A}+_{G_n}\mathcal{A}=\{a_i+a_j:(i,j)\in E(G_n)\},$$

and

$$\mathcal{A}\cdot_{G_n}\mathcal{A}=\{a_i a_j:(i,j)\in E(G_n)\}.$$

Strong Erdős–Szemerédi conjecture. For every $c>0$ and $\varepsilon>0$, there is a threshold $n_0$ such that, if $n\geq n_0$, then for every $n$-element set of reals $\mathcal{A}$ and every graph $G_n$ with at least $n^{1+c}$ edges,

$$|\mathcal{A}+_{G_n}\mathcal{A}|+|\mathcal{A}\cdot_{G_n}\mathcal{A}|\geq |\mathcal{A}|^{1+c-\varepsilon}.$$

This conjecture was refuted by Alon, Ruzsa, and Solymosi, so it is no longer open.