Weak BAB conjecture for Mori fiber spaces

Let $X\to T$ be a Mori fiber structure when $X$ is $\mathbb{Q}$-factorial with terminal singularities, the morphism is a contraction, $-K_X$ is ample over $T$, $\rho(X/T)=1$, and $\dim X>\dim T$. Let $X$ be of epsilon-Fano type if there is an effective $\mathbb{Q}$-divisor $\Delta$ such that $(X,\Delta)$ is an $\epsilon$-klt log Fano pair.

Weak BAB conjecture for Mori fiber spaces. Fix $0<\epsilon<1$ and an integer $n>0$. Then there exists a number $M(n,\epsilon)$ depending only on $n$ and $\epsilon$ such that, if $X$ is an $n$-dimensional variety of $\epsilon$-Fano type with a Mori fiber structure, then

$${\rm Vol}(-K_X)\leq M(n,\epsilon).$$

This is a restricted form of Weak BAB. The supplied text does not establish its status independently; the paper proves the unrestricted Weak BAB conjecture in dimension three, so the status of this restricted formulation should be checked.