Weak-BAB conjecture on boundedness of anti-canonical volumes

Fix a positive integer $n$ and a real number $\varepsilon>0$. Let $\mathfrak{D}$ be the set of all log pairs $(X,\Delta)$ such that $X$ is a projective variety of dimension $n$, $(X,\Delta)$ has $\varepsilon$-log canonical singularities for some boundary $\mathbb{R}$-divisor $\Delta$, and $-(K_X+\Delta)$ is ample. Weak-BAB conjecture. There exists a positive real number $M(n,\varepsilon)$, depending only on $n$ and $\varepsilon$, such that

$$\{\operatorname{vol}_X(-K_X)\mid (X,\Delta)\in\mathfrak{D}\text{ for some }\Delta\}$$

is bounded above by $M(n,\varepsilon)$.

This volume-boundedness statement is a necessary consequence of the BAB conjecture. It is proved in characteristic zero in every dimension as a consequence of Birkar's proof of BAB, while the positive-characteristic situation considered in the paper is only established in the stated three-dimensional setting.