Erdős–Sárközy–Stewart quadratic prime-factor conjecture for subset sums

For a finite set $A\ne\textstyle\varnothing$ of positive integers, let $\Sigma^*(A)$ denote the set of nonempty subset sums, and let $P(A)$ be the greatest prime factor of $\prod_{a\in A}a$. Erdős–Sárközy–Stewart's quadratic conjecture. There is an absolute constant $c>0$ such that

$$P(\Sigma^*(A))>c|A|^2.$$

This is a quantitative strengthening of the preceding prime-factor growth conjecture and is presented by the source as the second conjecture of Erdős, Sárközy, and Stewart.