McKernan's epsilon-delta conjecture for Mori fiber spaces

Let $m,n$ be fixed positive integers and let $\epsilon>0$. A variety is $\epsilon$-log terminal if all its log discrepancies are greater than $\epsilon$. A Mori fiber space is a projective morphism $f\colon X\to Y$ such that $-K_X$ is $f$-ample and $\rho(X/Y)=1$.

McKernan's conjecture. There exists a number $\delta=\delta_{m,n}(\epsilon)>0$ such that, whenever $X$ is a $\mathbb Q$-factorial variety, $f\colon X\to Y$ is a Mori fiber space with $\dim Y=n$ and $\dim X=m+n$, and $X$ is $\epsilon$-log terminal, the variety $Y$ is $\delta$-log terminal.

The conjecture predicts a uniform lower bound for the singularities of the base of Mori fiber spaces in fixed dimensions. The paper proves it in the toric case; the general conjecture is presented as open in the source.