Resolution of non-singularities and the absolute anabelian conjecture

Given two hyperbolic curves over p-adic local fields, the absolute anabelian conjecture claims that any isomorphism between their étale fundamental group comes from an isomorphism of schemes. This conjecture was proven by S. Mochizuki for curves of quasi-Belyi type. The goal of this paper is to extend the result to a wider family of curves, that includes for example hyperbolic Mumford curves.