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 $pi_*alpha = 0$. A conjecture of Debarre, Jiang, and Voisin predicts that $alpha$ 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.