A quasivariety lattice of torsion-free soluble groups

Let $L_q(qG)$ be a lattice of quasivarieties contained in a quasivariety generated by a group $G$. It is proved that if $G$ is a torsion-free finitely generated group in $\mathcal{AB}_{p^k}$, where $p$ is a prime, $p\ne2$, and $k\in\mathbf N$, which is a split extension of an Abelian group by a cyclic group, then the lattice $L_q(qG)$ is a finite chain.