The complexity of quasivariety lattices. II

We prove that if a quasivariety $\mathbf{K}$ contains a finite $\mathrm{B}^\ast$-class relative to some subquasivariety and some variety possessing some additional property, then $\mathbf{K}$ contains continuum many $Q$-universal non-profinite subquasivarieties having an independent quasi-equational basis as well as continuum many $Q$-universal non-profinite subquasivarieties having no such basis.