Minimal models from nef positive parts of Nakayama–Zariski decompositions

Let $(X,B+M)$ be a projective g-lc pair. Suppose there is a birational model $\pi\colon X'\to X$ such that the positive part $P_\sigma(\pi^*(K_X+B+M))$ is nef.

Minimal-model conjecture. Then $(X,B+M)$ has a minimal model.

This asks whether the existence of a birational Nakayama–Zariski decomposition with nef positive part characterises the existence of a minimal model for projective generalised log-canonical pairs. The source presents this as a converse to the known implication from the existence of a minimal model to such a decomposition; no resolution is supplied here.