The property of being equationally Noetherian for some soluble groups

Let $\mathfrak B$ be a class of groups $A$ which are soluble, equationally Noetherian, and have a central series
$$ A=A_1\geqslant A_2 \geqslant\ldots A_n\geqslant\ldots $$
such that $\bigcap A_n=1$ and all factors $A_n/A_{n+1}$ are torsion-free groups; $D$ is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that the wreath product $D\wr A$ is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary derived lengths and ranks are equationally Noetherian.