On dominions of the rationals in nilpotent groups

The dominion of a subgroup $H$ of a group $G$ in a class $M$ is the set of all $a\in G$ that have the same images under every pair of homomorphisms, coinciding on $H$ from $G$ to a group in $M$. A group $H$ is $n$-closed in $M$ if for every group $G=\operatorname{gr}(H,a_1,\dots,a_n)$ in $M$ that includes $H$ 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 rationals is $2$-closed in every quasivariety of torsion-free nilpotent groups of class at most $3$.