Numerical nonvanishing conjecture for generalized polarized pairs

Let $(X,B+\mathbf{M})$ be a projective generalized log canonical pair. Suppose that

$$K_X+B+\mathbf{M}_X$$

is pseudo-effective, and that

$$\mathbf{M}=\sum_j\mu_j\mathbf{M}_j,$$

where each $\mu_j\in\mathbb{R}_{>0}$ and each $\mathbf{M}_j$ is a nef Cartier $b$-divisor. Here, $K_X+B+\mathbf{M}_X$ is numerically equivalent to a divisor when their difference has zero intersection with every curve.

Numerical nonvanishing conjecture. There exists an effective $\mathbb{R}$-Cartier $\mathbb{R}$-divisor $D$ such that

$$K_X+B+\mathbf{M}_X\equiv D.$$

This conjecture concerns numerical effectivity for generalized polarized pairs and is used in the paper as an important problem related to the singular Campana–Peternell question. Its general status is not resolved in the supplied text.