Filip–Tosatti's non-semipositivity conjecture for the line bundle on Grauert's example

Let $L$ be the line bundle in Grauert's example, a projective surface example discussed by Filip and Tosatti. Filip–Tosatti's conjecture. The line bundle $L$ is nef and big, although it is not semipositive. The conjecture asks whether a nef and big line bundle on a projective surface can fail to be semipositive. The paper proves this claim for Grauert's example by explicitly computing its first obstruction class, so the conjecture is resolved.