Xiao–Reid's atomic-fiber Morsification conjecture

Let $f\colon S\to (p\in B)$ be a relatively minimal fiber germ of genus $g$. A splitting family of $f^{-1}(p)$ is a one-parameter family of relatively minimal fibrations with central fiber germ isomorphic to $f$; it is proper if every sufficiently nearby noncentral fiber has at least two singular fibers. The fiber germ $f^{-1}(p)$ is atomic when it has no proper splitting family.

Xiao–Reid's Morsification conjecture. A fiber germ $f^{-1}(p)$ is atomic if and only if it is a stable curve with one node or a multiple of a smooth curve.

This conjecture characterizes the fiber germs that cannot be split into simpler singular fibers and is the fiberwise form of the Morsification problem used in the paper's discussion of non-reduced fibers. The source presents it as a famous conjecture due to Xiao and Reid and gives no resolution.