Lazić–Peternell's generalized nonvanishing conjecture

Let $(X,\Delta)$ be a klt pair on a normal projective variety $X$ such that $K_X+\Delta$ is pseudo-effective. Let $L$ be a nef $\mathbb{Q}$-divisor on $X$. Generalized Nonvanishing Conjecture. For every rational number $t\geq0$, there exists an effective $\mathbb{Q}$-Cartier divisor $D$ such that

$$K_X+\Delta+tL\equiv D.$$

Equivalently, the numerical class of $K_X+\Delta+tL$ belongs to the effective cone. This conjecture is related to generalized pairs and the canonical bundle formula; the source does not state a resolution.