Koike's local unitary-flatness conjecture for semi-positive line bundles

Let $X$ be a complex manifold, let $L$ be a semi-positive holomorphic line bundle on $X$, and let $Y$ be a compact Kähler submanifold such that $L|_Y$ is topologically trivial. Let $V$ be a sufficiently small tubular neighborhood of $Y$ in $X$. Koike's conjecture. The restriction $L|_V$ is unitary flat.

The conjecture asks whether semi-positivity and topological triviality along $Y$ force the line bundle to become unitary flat on a neighborhood of $Y$. The paper investigates obstructions to this local linearization and applies the resulting analysis to construct nef and big line bundles that are not semi-positive.