Birational weak Zariski decomposition conjecture for generalized polarized pairs

Let $(X/Z,B+M)$ be a $Q$-factorial NQC g-dlt pair. If $K_X+B+M$ is pseudo-effective over $Z$, a birational NQC weak Zariski decomposition over $Z$ means that there is a birational morphism $g:Y\to X/Z$ such that

$$g^*(K_X+B+M)\equiv P+N/Z,$$

where $N\geq 0$ and $P$ is NQC over $Z$.

Birational weak Zariski decomposition conjecture. Every such pair admits a birational NQC weak Zariski decomposition over $Z$.

Weak Zariski decompositions are weaker than ordinary Zariski decompositions and are closely related to the existence of generalized log terminal models. The conjecture remains open in general.