Threefold product-of-sums growth conjecture

Let $A\subset\mathbb{R}$ be a finite set. Write

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

Threefold product-of-sums conjecture. For every $\epsilon>0$,

$$|(A+A)(A+A)(A+A)|\gg |A|^{3-\epsilon}.$$

The source presents this as a stronger estimate using fewer variables than the preceding bound, which includes a threefold sum of products and is bounded below by a quantity of order $|A|^3/\log|A|$. The conjecture remains unresolved in the source.