Fano-type base conjecture for Mori fibrations

Let $n>0$ be an integer and let $0<\epsilon<1$. A normal projective variety is of $\epsilon$-Fano type if there exists an effective $\mathbb{Q}$-divisor $B$ such that $(X,B)$ is an $\epsilon$-klt log Fano pair. Fano-type base conjecture. There exists a number $\delta(n,\epsilon)>0$ depending only on $n$ and $\epsilon$ such that, if $X$ is an $n$-dimensional variety of $\epsilon$-Fano type with a Mori fibration $X\to Z$, then $Z$ is of $\delta(n,\epsilon)$-Fano type. This predicts uniform control of the bases produced by Mori fibrations and is intended as an inductive tool in the minimal model program. Its general validity is open.