Lifting problem for minimally wild covers of Berkovich curves

Let k be an algebraically closed, complete non-Archimedean field.
I will briefly describe Berkovich spaces, skeletons of Berkovich curves, skeletons of morphisms of curves and various enhancements of skeletons.

The semistable reduction theorem asserts that every nice curve possesses a skeleton.
Furthermore, the simultaneous semistable reduction theorem asserts that any finite generically etale morphism of nice compact curves possesses a skeleton. We are interested in the inverse direction: given a finite morphism of graphs that can arise as skeletons, can it be lifted to a morphism of nice compact curves?

In general, the answer is no. However, enhancing the graphs to metric graphs with reduction k-curves attached to the vertices changes the answer: a lifting theorem of Amini-Baker-Brugallé-Rabinoff (2015) shows that any suitable morphism of such graphs lifts to a finite (generically etale) residually tame morphism of Berkovich curves.

In a joint work with Temkin, we introduce a new enhancement of the skeleton and provide it with new invariants that are trivial in the residually tame case. In this setting, we were able to generalize the lifting result to minimally residually wild morphisms.