Extension theorem without the deformable pseudoeffectivity hypothesis

Let $X$ be a compact Kähler manifold, let $H\to X$ be a holomorphic line bundle, and let $Z\subset X$ be a smooth hypersurface. Let $h$ be a singular Hermitian metric on $H$ satisfying the hypotheses in Theorem 2, except for the condition that $Z$ is deformably pseudoeffective. Extension conjecture. The conclusion of Theorem 2 holds even when the words “deformably pseudoeffective” are removed from its second line. The conjecture proposes removing the deformable pseudoeffectivity assumption from the authors’ $L^2$ extension theorem; the paper presents this as an open direction motivated by the lack of an adaptation of the Berndtsson–Lempert method to general hypersurfaces.