Iterated difference-product growth conjecture

Let $A\subset\mathbb{R}$ be a finite set, and for a positive integer $k$ write

$$(A-A)^{(k)}=\underbrace{(A-A)\cdots(A-A)}_{k\text{ factors}}.$$

Here $X\gg_{k,l}Y$ means that $X\geq c_{k,l}Y$ for a positive constant depending only on $k$ and $l$. Iterated difference-product growth conjecture. For every $l>0$ there exists $k>0$ such that, uniformly for all sets $A\subset\mathbb{R}$,

$$|(A-A)^{(k)}|\gg_{k,l}|A|^l.$$

This conjecture predicts arbitrarily large polynomial growth from sufficiently many iterated products of a difference set. The paper notes that even the case $l=3$ appears beyond the methods developed there; no resolution is supplied.