On full residuance of some groups

In this paper we define the rank of a finitely generated torsion free nilpotent group. The main result is the following
Theorem. Let $G$ be a finitely generated nilpotent group. Let $\mathfrak U$ be an arbitrary variety of groups. Assume that $G$ is torsion free, $\operatorname{rk}G=k$, $\mathfrak N:=\operatorname{var}G$, $G\cong F_k/R$, $R\triangleleft F_k$. Then for all $s>k$, the groups $F_s(\mathfrak{UN})$ are fully residual $F_k/U(R)$-groups.