Prokhorov–Shokurov conjectures on the moduli part of the canonical bundle formula

Let $f\colon(X,B)\rightarrow Z$ be an lc-trivial fibration, and let $M_Z$ denote the moduli divisor in the canonical bundle formula. For a birational model $Z'\rightarrow Z$, write $M_{Z'}$ for the corresponding birational transform of the moduli divisor.

Prokhorov–Shokurov conjectures. The following properties are conjectured:

1. Log canonical adjunction: There exists a proper birational morphism $Z'\rightarrow Z$ such that $M_{Z'}$ is semiample.
2. Particular case of effective log abundance: If $X_{\eta}$ is the generic fibre of $f$, then

$$I_0(K_{X_{\eta}}+B_{\eta})\sim 0,$$

where $I_0$ depends only on $\dim X_{\eta}$ and the multiplicities of the horizontal part of $B$. 3. Effective adjunction: There exists a positive integer $I$, depending only on the dimension of $X$ and the horizontal multiplicities of $B$, such that $IM_Z$ is the pullback of a divisor $M$ that is base point free on some model $Z'\rightarrow Z$.

These conjectures concern semiampleness and effective boundedness of the moduli part in the canonical bundle formula. The source presents them as conjectural properties; no resolution is supplied here.