The pseudo-effective pushforward conjecture for cohomology classes

Let $X$ and $Y$ be smooth complex projective varieties and let $\varphi:X\rightarrow Y$ be a morphism. For a smooth variety, pushforward in cohomology is the Gysin morphism obtained using Poincare duality. A subvariety of codimension $l$ is contracted by $\varphi$ exactly when its cohomology class is annihilated by this morphism.

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

This is the cohomological formulation of the preceding conjecture under Poincare duality. The source presents it as a conjectural extension of the established relationship for algebraic subvarieties; the general assertion remains open.