On the residual polynilpotence of tree products of groups with locally cyclic edge subgroups

Let $\mathcal{N}$ be a non-empty class of nilpotent groups closed under taking subgroups, quotient groups, finite direct products, and unrestricted direct powers of the form $\prod_{x \in X}Y_{x}$, where $X \in \mathcal{N}$ and $Y_{x}$ ($x \in X$) are isomorphic copies of a group $Y \in \mathcal{N}$