Balog–Roche-Newton–Zhelezov growth conjecture for difference sets

Let $A$ be a set of scalars with at least three elements, and put $D=A-A$. For a positive integer $m$, write $D^m$ for the $m$-fold product set of $D$. Balog–Roche-Newton–Zhelezov growth conjecture. For every positive integer $n$, there is a universally and polynomially related positive integer $m$ such that

$$|D^m|\geq |D|^n.$$

This question was raised by Balog, Roche-Newton and Zhelezov in connection with multiplicative growth of difference sets. The source presents it as an open question and does not state a resolution.