On the restriction of the moduli part to a reduced divisor

Let $f \colon (X,Δ) \to Y$ be a fibration such that $K_X + Δ$ is torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$ has dimension 3 and the fibres have dimension at most 3. Then the restriction of the moduli part to its augmented base locus is semiample.

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