A criterion for the $\mathscr F_\pi$-residuality of free products with amalgamated cyclic subgroup of nilpotent groups of finite ranks

Let $G$ be the free product of nilpotent groups $A$ and $B$ of finite rank with amalgamated cyclic subgroup $H$, $H\ne A$ and $H\ne B$. Suppose that, for some set $\pi$ of primes, the groups $A$ and $B$ are residually $\mathscr F_\pi$, where $\mathscr F_\pi$ is the class of all finite $\pi$-groups. We prove that $G$ is residually $\mathscr F_\pi$ if and only if $H$ is $\mathscr F_\pi$-separable in $A$ and $B$.