Effective b-semiampleness conjecture

Let $d$ be a nonnegative integer and let $\Gamma\subset [0,1]$ be a rational dcc subset. For a $0$-contraction $(X,D)\to Z$ under the assumptions cited in the source, write $\mathcal D_{\rm mod}$ for its moduli b-divisor. A b-divisor is b-free if it is the closure $\overline M$ of a base point free divisor $M$ on some model $Z'$ of $Z$.

Effective b-semiampleness conjecture. There exists a positive integer $I=I(d,\Gamma)$ such that, for every such $0$-contraction with $\dim X=d$ and $D^{\mathrm h}\in\Gamma$, $I\mathcal D_{\rm mod}$ is b-free:

$$I\mathcal D_{\rm mod}=\overline M.$$

Here $M$ is a base point free divisor on some model $Z'$ of $Z$. Equivalently, $(Z,D_{\rm \div}+\mathcal D_{\rm mod})$ is an Alexeev log pair of index $I$. The same is expected for $0$-contractions of Alexeev pairs of index $m$, with $I(d,\Gamma,m)$ depending also on $m$.

This conjecture would provide effective semiampleness for the moduli part in the adjunction framework and is intended to support reductions to finite-type and Calabi–Yau cases. The source attributes it to PSh08 and leaves it open.