The log abundance conjecture for log canonical pairs

Let $(X,\Delta)$ be a log pair, with $X$ normal and $\Delta$ an effective formal $\mathbb{Q}$-linear combination of prime divisors. The pair is log canonical if all discrepancies are at least $-1$; a divisor is semi-ample if some positive multiple is base-point-free. The log abundance conjecture. If $(X,\Delta)$ is log canonical and $\Delta$ is effective, then $K_X+\Delta$ is nef if and only if it is semi-ample.

The paper discusses abundance for pairs on blow-ups of projective space and says it proves the relevant abundance statement for the constructed family, while the general conjecture is a central question of the minimal model program.