$\omega$-Independent bases for quasivarieites of torsion-free groups

It is proved that there exists a set $\mathcal{R}$ of quasivarieties of torsion-free groups which (a) have an $\omega$-independent basis of quasi-identities in the class $\mathcal{K}_{0}$ of torsion-free groups, (b) do not have an independent basis of quasi-identities in $\mathcal{K}_{0}$, and (c) the intersection of all quasivarieties in $\mathcal{R}$ has an independent quasi-identity basis in $\mathcal{K}_{0}$. The collection of such sets $\mathcal{R}$ has the cardinality of the continuum.