Uniform boundedness conjecture for log-pluricanonical maps

Let $n$ be a positive integer and let $I\subseteq [0,1]\cap\mathbb{Q}$ be a DCC set. Let $\mathfrak{D}$ be a collection of log pairs such that: $X$ is a projective variety of dimension $n$ over an algebraically closed field; $(X,\Delta)$ is log canonical and the coefficients of $\Delta$ belong to $I$; and $K_X+\Delta$ is big.

Uniform boundedness conjecture. There is a positive integer $N=N(n,I)$ depending only on $n$ and $I$ such that the linear system

$$|\lfloor m(K_X+\Delta)\rfloor|$$

defines a birational map onto its image for every $m>N$ and every $(X,\Delta)\in\mathfrak{D}$.

This is a uniform boundedness statement for log-pluricanonical maps. The assertion is proved in characteristic $0$ in all higher dimensions by Hacon, McKernan and Xu as part of their inductive arguments for the ACC property of log canonical thresholds; the positive-characteristic case is the subject of the paper and is resolved by the stated result.