A relatively finite-to-finite universal but not Q-universal quasivariety

It was proved by the authors that the quasivariety of quasi-Stone algebras \(\mathbf {Q}_{\mathbf {1,2}}\) is finite-to-finite universal relative to the quasivariety \(\mathbf {Q}_{\mathbf {2,1}}\) contained in \(\mathbf {Q}_{\mathbf {1,2}}\). In this paper, we prove that \(\mathbf {Q}_{\mathbf {1,2}}\) is not Q-universal. This provides a positive answer to the following long standing open question: Is there a quasivariety that is relatively finite-to-finite universal but is not Q-universal?