Viehweg and Zuo obtained several results concerning the moduli number in smooth families of polarized varieties with semi-ample canonical class over a quasiprojective base. These results led Viehweg to conjecture that the base of a family of maximal variation is of log-general type, and the conjecture has been recently proved by Campana and Paun. From the “opposite” side, Taji proved that a smooth projective family over a special (in the sense of Campana) quasiprojective base is isotrivial. We extend Taji’s theorem to quasismooth families, that is, families of leaves of compact foliations without singularities.