  It is conjectured that the canonical models of varieties (not of general
type) are bounded when the Iitaka volume is fixed. We confirm this conjecture
when the general fibers of the corresponding Iitaka fibration are in a fixed
bounded family of polarized Calabi-Yau pairs. As a consequence, we prove that
in this case, the fibration is birationally bounded, and when it has terminal
singularities, the corresponding minimal model is bounded in codimension 1.
