Shokurov's conjecture on singularities of bases of Fano type fibrations

Assume $d\in\mathbb N$ and $\epsilon\in\mathbb R^{>0}$. An $\epsilon$-lc pair $(X,B)$ has log discrepancies at least $\epsilon$; let $f\colon X\to Z$ be a contraction with $\dim Z>0$ such that $K_X+B\sim_{\mathbb R}0/Z$ and $-K_X$ is big over $Z$.

Shokurov's conjecture. There is $\delta\in\mathbb R^{>0}$ such that one can write

$$K_X+B\sim_{\mathbb R}f^*(K_Z+B_Z+M_Z)$$

such that $(Z,B_Z+M_Z)$ is $\delta$-lc, where $B_Z$ and $M_Z$ are the discriminant and moduli parts of adjunction.

The conjecture is known when $\dim X-\dim Z\leq 1$ and when the general fibre together with the restricted boundary belongs to a bounded family. It remains open in higher dimension without that boundedness condition.