Weak BAB conjecture for anti-canonical volumes

Let $X$ be a variety of epsilon-Fano type if there is an effective $\mathbb{Q}$-divisor $\Delta$ such that $(X,\Delta)$ is an $\epsilon$-klt log Fano pair. For a divisor, write ${\rm Vol}(-K_X)$ for its volume.

Weak BAB conjecture. 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, then

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

This volume-boundedness statement is a consequence of the BAB conjecture and is the weaker problem studied in the paper. It is stated as open in general dimension three or higher.