Morphisms and faces of pseudo-effective cones

Let $\pi :X\to Y$ be a morphism of projective varieties and suppose that $\alpha$ is a pseudo-effective numerical cycle class satisfying $\pi_*\alpha = 0$\mathrm{.}AconjectureofDebarre,Jiang,andVoisinpredictsthat$\alpha$ is a limit of classes of effective cycles contracted by $\pi$. We establish new cases of the conjecture for higher codimension cycles. In particular we prove a strong version when $X$ is a fourfold and $\pi$ has relative dimension one.