Campana's isotriviality conjecture for polarized families with semi-ample canonical bundle

Let $f_U:U\to V$ be a smooth projective family, and suppose that $V$ has a morphism $\mu:V\to P_h$, where $P_h$ is the coarse moduli scheme associated with the moduli functor of polarized projective manifolds with semi-ample canonical bundle and fixed Hilbert polynomial $h$. Assume that $V$ is special. Campana's generalized isotriviality conjecture. The family $f_U$ is isotrivial. This extends Campana's conjecture from canonically polarized families to polarized families with semi-ample canonical bundle and fixed Hilbert polynomial.