On a question of Demailly-Peternell-Schneider

This note aims to give an affirmative answer to an open question posed by Demailly-Peternell-Schneider [DPS] in 2001 and recently by Peternell [P] again. Let $f:X\mapsto Y$ be a surjective morphism from a log canonical pair (X,D) onto a $\mathbb{Q}$-Gorenstein variety $Y$. If $-(K_X+D)$ is nef, we show that $-K_Y$ is pseudo-effective.