Generic invariance of Fano type in families

Let $X\to S$ be a projective and surjective morphism between normal varieties over $\a0\mathbb{C}$, and let $S'\subseteq S$ be a Zariski-dense subset such that for every closed point $s\in S'$, the fiber $X_s$ is of Fano type. If $\a0\eta\in S$ is the generic point, write $X_{\overline{\eta}}$ for the geometric generic fiber.

Generic invariance conjecture. The geometric generic fiber $X_{\overline{\eta}}$ is of Fano type.

This is a characteristic-zero generic-invariance statement for Fano type in a projective family. The paper proves related results when anti-canonical volumes are constant on a Zariski-dense set and when the Fano type fibers are surfaces; the unrestricted assertion remains open in the source.