  In this paper, we proved that a log smooth family of log general type klt
pairs with a special (in the sense of Campana) quasi-projective base is
isotrivial. As a consequence, we proved the generalized Kebekus-Kov\'acs
conjecture \cite[Conjecture 1.1]{WW19}, for smooth families of general type
varieties as well as log smooth families of log canonical pairs of log general
type, assuming the existence of relative good minimal models.
