Fundamental properties of basic slc-trivial fibrations, II

We prove that if the moduli $\mathbb{Q}$-b-divisor of a basic slc-trivial fibration is b-numerically trivial then it is $\mathbb{Q}$-b-linearly trivial. As a consequence, we prove that the moduli part of a basic slc-trivial fibration is semi-ample when the base space is a curve.