Extent to which the log different governs the wild locus

Determine to what extent the log different governs the shape of the wild locus of an étale morphism of adic spaces, extending the known descriptions for Berkovich curves over algebraically closed fields.

Background

The paper proves that the wild locus of an étale morphism of smooth adic spaces is either empty or contains a divisorial point, using the wild degree as the principal invariant. The authors note that a more refined invariant, the log different, is expected to provide substantially sharper local information about wildness.

For Berkovich curves over algebraically closed fields, the different gives a radial description of the wild locus. The paper explicitly leaves unresolved how far an analogous control by the log different extends in the broader setting of adic spaces and non-algebraically closed or higher-rank base fields; it also notes that some wild extensions are not detected by the log different.