Existence of Independent Quasi-Equational Bases

We give a sufficient condition for a quasivariety K, weaker than the one found earlier by A. V. Kravchenko, A. M. Nurakunov, and the author, which ensures that K contains continuum many subquasivarieties with no independent quasi-equational basis relative to K. This condition holds, in particular, for any almost f f-universal quasivariety K.