Sum-product exponent conjecture for positive integers

Let $A$ be a set of $n\ge 2$ positive integers, and define

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

Sum-product exponent conjecture. Then

$$|A+A|+|AA|\ge |A|^{(1+\sqrt5)/2}.$$

This conjecture proposes a quantitative lower bound on the combined additive and multiplicative expansion of a finite set. The paper presents it as one of several dataset-motivated conjectures, with no resolution supplied.