On $2$-closedness of the rational numbers in quasivarieties of nilpotent groups

The dominion of a subgroup $H$ of a group $G$ in a class $M$ is the set of all elements $a\in G$ that have equal images under every pair of homomorphisms from $G$ to a group of $M$ coinciding on $H$. A group $H$ is said to be $n$-closed in $M$ if for every group $G=\operatorname{gr}(H,a_1,\dots,a_n)$ of $M$ that contains $H$ and is generated modulo $H$ by some $n$ elements, the dominion of $H$ in $G$ (in $M$) is equal to $H$. We prove that the additive group of the rational numbers is $2$-closed in every quasivariety $M$ of torsion-free nilpotent groups of class at most $3$ whenever every $2$-generated group of $M$ is relatively free.