  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.
