Bobadilla–Kollár's homology-fiber-bundle smoothness conjecture

Let $f:X\to Y$ be a proper morphism between complex analytic spaces, with $X$ and $Y$ smooth. For a commutative ring $R$, call $f$ an $R$-homology fiber bundle if $Y$ has an open cover $Y=\bigcup_iU_i$ such that, for every $i$ and every $y\in U_i$, the inclusion

$$H_*(f^{-1}(y),R)\hookrightarrow H_*(f^{-1}(U_i),R)$$

is an isomorphism.

Bobadilla–Kollár's conjecture. If $f$ is a $\mathbb Z$-homology fiber bundle, then $f$ is smooth.

This conjecture links local topological constancy of the homology of fibers with analytic smoothness. The paper uses it to deduce the stronger Kotschick conjecture in certain cases, while the general conjecture remains open in the supplied text.