Finiteness of a set of quasivarieties of torsion-free metabelian groups of axiomatic rank 2

Let $\mathcal M$ be a quasivariety of all torsion-free groups in which squares of elements are commuting. It is proved that the set of quasivarieties contained in $\mathcal M$ and defined by quasi-identities in two variables is finite.