Small-representative conjecture for sum-product pairs

Let

$$\operatorname{SPP}(n)=\{(|A+A|,|AA|):A\subseteq\mathbb N,\ |A|=n\},$$

where

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

Small-representative conjecture. If $(i,j)\in\operatorname{SPP}(n)$, then there is a set $A\subseteq\mathbb N$ with

$$|A|=n,\qquad |A+A|=i,\qquad |AA|=j,$$

and

$$\max A\le 2^{\left\lfloor(3n-4)/2\right\rfloor}.$$

This conjecture bounds the largest element needed to realize any sum-product pair of cardinality $n$. The source motivates the bound from its computational dataset and gives no proof or disproof.