The b-semiampleness conjecture for klt-trivial fibrations

Let $f\colon(X,B)\rightarrow Z$ be a klt-trivial fibration, meaning that $(X,B)$ is Kawamata log terminal at the generic point of $Z$ and $K_X+B\sim_{\mathbb{Q}f}0$. The canonical bundle formula writes

$$K_X+B\sim_{\mathbb{Q}}f^{\ast}(K_Z+B_Z+M_Z),$$

where $B_Z$ is the discriminant divisor and $M_Z$ is the moduli part. The b-semiampleness conjecture. There exists a birational morphism $\mu\colon Z'\rightarrow Z$ such that $M_{Z'}$ is semiample.

This is the analogous b-semiampleness assertion for klt-trivial fibrations, and the source refers to it as a known formulation in the literature. The source provides no resolution evidence, so its database status is open.