Erdős's convex sumset conjecture

Let $A\subset\mathbb{R}$ be a finite convex set, meaning that if $A=\{a_1<a_2<\cdots<a_k\}$, then

$$a_i-a_{i-1}<a_{i+1}-a_i$$

for every $1<i<k$. Erdős's conjecture. For every $\epsilon>0$,

$$|A+A|\gg_{\epsilon}|A|^{2-\epsilon}.$$

This conjecture concerns the additive growth of convex sets. The paper proves a related lower bound for sums with an arbitrary finite set, but the stated conjecture remains open.