The non-vanishing conjecture for projective lc pairs

Let $(X,\Delta)$ be a projective lc pair, meaning that $X$ is a projective normal variety and $(X,\Delta)$ has log canonical singularities. Assume that $K_X+\Delta$ is pseudo-effective.

Non-vanishing conjecture. There exists an effective $\mathbb{R}$-divisor $E\geq 0$ such that

$$K_X+\Delta\sim_{\mathbb{R}}E.$$

This is one of the most important open problems in minimal model theory. The paper proves it when the underlying variety $X$ is of Calabi--Yau type, namely when there is an effective $\mathbb{R}$-divisor $C$ such that $(X,C)$ is log canonical and $K_X+C\equiv 0$.