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In this paper, we show the semi-ampleness of an arbitrarily small perturbation of the moduli b-divisor by a fixed appropriate divisor which roughly speaking comes from a section of $K_X+B$.\n  We apply the above result to settle a conjecture of Fujino and Gongyo: if $f\\colon X\\to Z$ is a smooth surjective morphism of smooth projective varieties with $-K_X$ semi-ample, then $-K_Z$ is also semi-ample. 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