Effectivity conjecture for negative parts of Zariski decompositions

Let $X$ be a projective variety, let $\alpha$ be a cycle class admitting a Zariski decomposition, and write

$$\alpha=P(\alpha)+N(\alpha).$$

Here $P(\alpha)$ and $N(\alpha)$ are respectively the positive and negative parts. Effectivity conjecture for negative parts. There is a proper closed subscheme $i:Y\subsetneq X$ and a pseudo-effective class $\beta\in N_k(Y)$ such that

$$N(\alpha)=i_*\beta.$$

The conjecture predicts that negative parts are supported on proper subvarieties; in the divisor case, it predicts effectiveness of the negative part. Its status is not resolved in the supplied text.