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Assuming the LMMP and the existence of log resolutions in dimension $\\leq n$, we prove that, when $K_X+B$ is $f$-nef, the moduli part is nef up to a birational map $Y \\dashrightarrow X$. As a corollary, we prove positivity of the moduli part in the $K$-trivial case, i.e. when $K_X+B \\sim_{\\Q} f^*L$ for some $\\Q$-Cartier $\\Q$-divisor $L$ on $Z$. 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