Viehweg hyperbolicity conjecture for KSB-stable Whitney equisingular families

Let $f:U\to V$ be a KSB-stable Whitney equisingular family, meaning a family of KSB-stable varieties with Whitney equisingular fibers. The family has maximal variation when the associated moduli map has image of dimension equal to $\dim V$. The base $V$ is of log general type when a smooth compactification has big logarithmic canonical divisor.

Viehweg hyperbolicity conjecture for KSB-stable families. If $f$ has maximal variation, then $V$ is of log general type.

This conjecture proposes that every Whitney equisingular stratum of the moduli space of KSB-stable varieties is hyperbolic in a birational-geometric sense. The paper establishes a first step for families with isolated rational hypersurface singularities, while the assertion for general KSB-stable Whitney equisingular families remains open.