Disproved. The status-defining source is Theorem 4 of Alon, Ruzsa and Solymosi [ARS20] (Publ. Mat. 64 (2020), 143--155, refereed): for every $0<c<1$ there is $\delta>0$ such that for infinitely many $n$ some set of $n$ integers carries a graph with at least $n^{1+c}/\log^{O(1)}n$ edges along which sums and products together number at most $n^{1+c-\delta}\log^{O(1)}n$ (the paper's $\Omega_l$ and $O_l$, loose up to powers of the logarithm, which the strict inequality in $\delta$ absorbs); their Theorem 3 is the explicit case with $\gg n^{5/3-o(1)}$ edges and $\max(\lvert A+_GA\rvert,\lvert A\cdot_GA\rvert)\ll n^{4/3+o(1)}$. The paper's positive result, that a graph with $m$ edges on $n$ integers has $\max(\lvert A+_GA\rvert,\lvert A\cdot_GA\rvert)\gg m^{3/2}n^{-7/4}$, bounds how far the failure can go. The claim page is Alon, Ruzsa and Solymosi (accepted on the refereed publication and the site's credit).