Semipositivity conjecture for divisors with topologically trivial normal bundle

Let $X$ be a complex surface and let $Y$ be a compact smooth curve holomorphically embedded in $X$ such that its normal bundle is topologically trivial. Let $[Y]$ denote the line bundle on $X$ corresponding to the divisor $Y$, and let a pair $(Y,X)$ be of type $(\beta)$ in the sense used in the paper.

Semipositivity conjecture. The line bundle $[Y]$ admits a $C^\infty$ Hermitian metric with semi-positive curvature if and only if the pair $(Y,X)$ is of type $(\beta)$.

The conjecture is motivated by the preceding corollary, which proves this equivalence under the additional hypotheses imposed there. The general claim concerns complex surfaces containing compact curves with topologically trivial normal bundle.