McKernan's conjecture on singularities of Mori fibre space bases

Assume $d\in\mathbb N$ and $\epsilon\in\mathbb R^{>0}$. A Mori fibre space is a contraction $f\colon X\to Z$ whose relative canonical divisor is anti-ample; an $\epsilon$-lc variety has log discrepancies at least $\epsilon$, and $\mathbb Q$-factorial means every Weil divisor has a suitable positive multiple that is Cartier.

McKernan's conjecture. There is $\delta\in\mathbb R^{>0}$ such that if $f\colon X\to Z$ is a Mori fibre space, where $X$ is an $\epsilon$-lc $\mathbb Q$-factorial variety of dimension $d$, then $Z$ is $\delta$-lc.

The conjecture generalises known results for threefold Mori fibre spaces and is known in the toric case. Its validity in higher dimension is presented as an open problem.