AbstractLet $$(X,\Delta )$$ ( X , Δ ) be a projective, $${{\mathbb {Q}}}$$ Q -factorial log canonical pair and let L be a pseudoeffective $${{\mathbb {Q}}}$$ Q -divisor on X such that $$K_X + \Delta + L$$ K X + Δ + L is pseudoeffective. Is there an effective $${{\mathbb {Q}}}$$ Q -divisor M on X such that $$K_X + \Delta + L$$ K X + Δ + 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.