The near-quadratic energy conjecture for convex sets

Let $A\subset\mathbb{R}$ be a finite convex set, meaning that if $A=\{a_1<\cdots<a_n\}$ then the consecutive differences $a_{i+1}-a_i$ are strictly increasing. Define the additive energy by

$$E(A)=\left|\{(a,b,c,d)\in A^4:a+b=c+d\}\right|.$$

Near-quadratic energy conjecture. If $A\subset\mathbb{R}$ is convex, then

$$E(A)\leq |A|^{2+o(1)}.$$

The trivial solutions already give $E(A)\geq |A|^2$, while the best-known general upper bound is $E(A)\ll |A|^{123/50+o(1)}$. No construction is known with energy at least $|A|^{2+c}$ for a fixed positive $c$, so the conjecture remains open.