A remark on generalized abundance for surfaces

Let (X, Delta) be a projective klt pair of dimension 2 and let L be a nef Cartier divisor on X such that K-X + Delta + L is nef. As a complement to the Generalized Abundance Conjecture by Lazic and Peternell, we prove that if K-X + Delta and L are not proportional modulo numerical equivalence, then K-X + Delta + L is semiample. An example due to Lazic shows that this is no longer true in any dimension n >= 3.