For any quasivariety \({\mathbf K}\) of algebras (of finite type), let \(L({\mathbf K})\) be the lattice of all quasivarieties in \({\mathbf K}\). Call \({\mathbf K}\) \(Q\)-universal iff for any quasivariety \({\mathbf M}\) (of algebras of finite type), \(L({\mathbf M})\) is a homomorphic image of a sublattice of \(L({\mathbf K})\). It follows, among other things, that \(L({\mathbf K})\) satisfies no nontrivial lattice identity whenever \({\mathbf K}\) is \(Q\)-universal. \textit{W. Dziobiak} [Algebra Univers. 21, 62-67 (1985; Zbl 0589.08007)] found a set of 4 conditions on a quasivariety \({\mathbf K}\) (of finite type) jointly ensuring that \(L({\mathbf K})\) fails to satisfy any nontrivial lattice identity. The main theorem of this paper states that Dziobiak's conditions are in fact sufficient to show that \(L({\mathbf K})\) is even \(Q\)-universal. As a corollary, the authors obtain that a quasivariety \({\mathbf K}\) (i) whose finite algebras carry no skew congruences and (ii) which contains an infinite family of finite hereditarily simple algebras none of which is embeddable into any other one, must be \(Q\)-universal.