Prokhorov–Shokurov's effective base-point-freeness conjecture

Let $d$ be a positive integer and let $\Gamma_0$ be a finite set of rational numbers. An lc-trivial fibration is a fibration $f:(X,B)\rightarrow Z$ with the usual lc-triviality condition, and $\mathbf{M}$ denotes its moduli part. Assume that

$$f:(X,B)\rightarrow Z$$

is an lc-trivial fibration such that $\dim X-\dim Z=d$ and the coefficients of the horizontal$/Z$ part of $B$ belong to $\Gamma_0$. Prokhorov–Shokurov's conjecture. There exists a positive integer $I$, depending only on $d$ and $\Gamma_0$, such that $I\mathbf{M}$ is base-point-free. The conjecture is known when $d=1$; its non-effective version is known for $d=2$, while the case $d\geq 3$ remains largely unresolved. It is important for the study of moduli spaces of varieties, especially log Calabi–Yau varieties.