Bobadilla–Kollár conjecture on projective morphisms over the disc

Let $Y$ be a complex analytic manifold and let

$$f \colon Y \longrightarrow \Delta$$

be a projective morphism to the complex disc. A homotopy fiber bundle is a morphism whose fibers are homotopy equivalent in the corresponding locally trivial sense, and a $\mathbb Z$-homology fiber bundle is one whose fibers are locally identified in integral homology. Bobadilla–Kollár conjecture. The following conditions are equivalent: (i) $f$ is submersive, hence a differentiable fiber bundle; (ii) $f$ is a homotopy fiber bundle; and (iii) $f$ is a $\mathbb Z$-homology fiber bundle. This conjecture gives a proposed topological characterization of submersive projective morphisms and is motivated by its role in special cases of Kotschick's conjecture on holomorphic one-forms without zeros. The source provides no evidence of a general resolution.