On a conjecture of Demailly-Peternell-Schneider: the Kahler case

Let $f:(X,\mathrm{\Delta })\to Y$ be a surjective holomorphic map between two normal Kahler varieties, where (X,Δ) is a log canonical pair, -(KX+Δ) is nef and Y is Q-Gorenstein. In Fu-Guo-Song-Wang [18], it is proved that -KY is pseudo-effective if f is a projective morphism. In this paper, prove that -KY is pseudo-effective without assuming that f is projective