The weakly pseudo-effective pushforward conjecture

Let $X$ and $Y$ be smooth complex projective varieties and let $\varphi:X\rightarrow Y$ be a morphism. A cohomology class is weakly pseudo-effective if it is a limit of classes of algebraic cycles represented by positive currents. Every pseudo-effective cycle class is weakly pseudo-effective, but the converse is known only in the divisor case.

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

This strengthens the weak, vector-space form of the cohomological pseudo-effective pushforward conjecture by replacing algebraic pseudo-effectivity with analytic weak pseudo-effectivity. The source notes that this conjecture would have strong consequences for the generalized Hodge conjecture, and its general validity remains open.