The nonvanishing conjecture for log canonical pairs

Let $(X/Z,B)$ be a log canonical (lc) pair, and suppose that $K_X+B$ is pseudo-effective over $Z$. A divisor $M$ is effective, written $M\ge 0$, when all its coefficients are nonnegative; $\mathbb R$-linear equivalence over $Z$ is denoted by $\sim_{\mathbb R}/Z$. Nonvanishing conjecture. There is an effective $\mathbb R$-divisor $M\ge 0$ such that

$$K_X+B\sim_{\mathbb R}M/Z.$$

This is known in dimension $3$, but very little is known in higher dimensions. The source describes it as a central problem and notes that it implies the minimal model conjecture.