Prokhorov–Shokurov's effective b-semiampleness conjecture

Fix an integer $d$. For every lc-trivial fibration $f\colon X\to Y$ of relative dimension $d$, choose a birational diagram

$$\begin{CD} (X',\Delta') @>{\phi}>> (X,\Delta)\\ @V{f'}VV @VV{f}V\\ Y' @>{\psi}>> Y. \end{CD}$$

Here $\phi$ and $\psi$ are birational, and $\mathbb M_{Y'}$ denotes the trace of the moduli b-divisor.

Effective b-semiampleness conjecture. There is an integer $I=I(d)$ such that one can choose the diagram with $K_{X'}+\Delta'=\phi^*(K_X+\Delta)$, $I\mathbb M_{Y'}$ Cartier and free, and ${\bf M}$ descending to $\mathbb M_{Y'}$.

This is the effective, uniform form of the b-semiampleness expectation: the index depends only on the relative dimension. The conjecture is open in the stated generality.