Solymosi–Volkmann sum-product inequality conjecture

Let $A$ be a set of $n$ positive integers, and define

$$A+A=\{a+b:a,b\in A\},\qquad AA=\{ab:a,b\in A\}.$$

Solymosi–Volkmann conjecture. One should have

$$|A+A|\,|AA|^2\ge \frac{n(n+1)}2(2n+1)^2,$$

with equality if and only if $A$ is a geometric progression. The source states that this conjecture is supported by the dataset for positive integers, but also notes that the inequality fails for some sets of positive real numbers. Thus the stated integer version is recorded as open.