The pseudo-effective pushforward conjecture for homology classes

Let $X$ and $Y$ be complex projective varieties and let $\varphi:X\rightarrow Y$ be a morphism. An irreducible subvariety of $X$ is contracted by $\varphi$ if its image has strictly smaller dimension. The pseudo-effective cone in $H_{2k}(X,\mathbf{R})$ is the closed convex cone generated by classes of irreducible subvarieties.

Pseudo-effective pushforward conjecture. Any pseudo-effective class $\alpha\in H_{2k}(X,\mathbf{R})$ such that $\varphi_*\alpha=0$ in $H_{2k}(Y,\mathbf{R})$ belongs to the real vector space, respectively the closed convex cone, spanned by classes of $k$-dimensional subvarieties of $X$ contracted by $\varphi$.

The closed-cone version is the stronger assertion, called the Strong Conjecture in the source, while the vector-space version is the Weak Conjecture. The paper proves these assertions for classes of curves and divisors and studies their implications for generalized Hodge-type statements; the general case remains open.