The neighborhood nefness conjecture for log canonical pairs

Let $\pi\colon X\to Y$ be a projective surjective morphism of normal complex varieties, let $(X, \Delta)$ be a log canonical pair, and let $W$ be a compact subset of $Y$. Assume that $K_X+\Delta$ is $\pi$-nef over $W$.

Neighborhood nefness conjecture. Then $K_X+\Delta$ is $\pi$-nef over some open neighborhood of $W$.

This conjecture asks whether relative nefness over a compact subset extends to a neighborhood of that subset. The source presents it as an important conjecture after explaining that its preceding results avoid the abundance conjecture for projective morphisms of complex analytic spaces; no resolution is supplied here.