Prokhorov–Shokurov conjecture on the b-semi-ampleness of moduli divisors

Let $X$ be a compact complex manifold with an lc-trivial contraction $\sigma\colon (X,R)\to Y$ over a smooth projective variety $Y$, where $R$ is a $\mathbb{Q}$-divisor on $X$. The contraction need not be a projective morphism. The discriminant $\mathbb{Q}$-divisor associated with $(\sigma,R)$ is denoted by $\Delta=\Delta(\sigma,R)$. Prokhorov–Shokurov's conjecture. One has

$$K_X+R\sim_{\mathbb{Q}}\sigma^*(K_Y+\Delta+M),$$

where the moduli $\mathbb{Q}$-divisor $M$ is $b$-semi-ample. This extends the canonical bundle formula to lc-trivial contractions in the compact complex, not necessarily projective, setting. The conjecture is attributed to Prokhorov and Shokurov and is presented here as applicable to this analytic setting; its resolution status is not specified in the source.