Conjecture on small sumsets of iterated product sets

Let $A$ be a finite set of real numbers, and let $k\geq 1$ be an integer. Define

$$A^{(k)}:=\{a_1\cdots a_k:a_1,\ldots,a_k\in A\}.$$

Iterated product-set sum conjecture. The $k$-fold sumset of $A^{(k)}$ should satisfy

$$|kA^{(k)}|\gg |A|^k.$$

The preceding theorem proves the weaker bound with $4^{k-1}$ summands instead of $k$ summands. The conjecture asks whether the number of summands can be reduced to $k$; the source gives no resolution.