The p-power base point free conjecture for klt pairs

Let $k$ be a field of characteristic $p>0$. Let $(X,\Delta)$ be a klt pair over $k$, and let $f:X\to Z$ be a projective $k$-morphism to a quasi-projective $k$-scheme $Z$. Let $L$ be an $f$-nef Cartier divisor on $X$ such that $L-(K_X+\Delta)$ is $f$-nef and $f$-big. The p-power base point free conjecture. Then $L$ is $p$-power $f$-free; that is, there exists a positive integer $e$ such that $p^eL$ is $f$-free. This is the positive-characteristic analogue of the Kawamata–Shokurov base point free theorem. The usual base point free statement fails in positive characteristic, while the cited examples motivate replacing ordinary freeness by $p$-power freeness; the conjecture's resolution is not established by the supplied context.