A question about generalized nonvanishing

Abstract

Let $(X,\Delta )$ be a projective, $\mathbb{Q}$-factorial log canonical pair and let $L$ be a pseudoeffective $\mathbb{Q}$-divisor on $X$ such that $K_X+\Delta +L$ is pseudoeffective. Is there an effective $\mathbb{Q}$-divisor $M$ on $X$ such that $K_X+\Delta +L$ is numerically equivalent to $M$? We are not aware of any counterexamples, but the answer is not completely clear even in the case of surfaces.

Final version, to appear in Rendiconti del Circolo Matematico di Palermo