The non-vanishing conjecture for projective log canonical pairs

Let $(X,\Delta)$ be a projective log canonical pair. If $K_{X}+\Delta$ is pseudo-effective, then there is an effective $\mathbb{R}$-divisor $D$ such that $K_{X}+\Delta \sim_{\mathbb{R}}D$.

Non-vanishing conjecture. The stated effective representative exists.

This is a central conjecture in the minimal model program. The conjecture is known in dimensions at most three, but remains open in higher dimension.