Generalization of the comprehensive factorization for comorphisms of sites

Determine whether the analog of the comprehensive (initial, discrete opfibration) factorization for comorphisms of sites, which was used to factor essential geometric morphisms in Caramello (2020), extends to the (terminally connected, pro-etale) framework for arbitrary geometric morphisms between Grothendieck topoi.

Background

The paper develops a (terminally connected, pro-etale) factorization for all geometric morphisms between Grothendieck topoi, generalizing known factorizations that apply in more restricted settings. The authors note several connections to the comprehensive factorization system for functors (initial, discrete opfibration).

In earlier work (Caramello, 2020), the factorization of essential geometric morphisms used an analog of the comprehensive factorization for comorphisms of sites. Here, the authors explicitly state uncertainty about whether that analog can be generalized to their broader setting involving pro-etale morphisms and terminally connected morphisms.