2-coherence of the higher container model

Prove that the higher container model of dependent type theory introduced by Altenkirch and Kaposi (2021) is 2-coherent—that is, show it satisfies the triangle and pentagon coherences for its wild category of contexts, has type triangulators and type pentagonators for substitution in types, and possesses coherators for context extension, thereby forming a structurally 2-coherent wild category with families.

Background

After defining structurally 2-coherent wild cwfs—requiring a 2-coherent wild category of contexts and additional coherence data for types and context extension—the authors verify these conditions for set-level cwfs and universe cwfs. They then posit that the higher container model should also meet these coherence requirements.

Establishing 2-coherence for the container model would extend the class of internal models captured by the framework and validate the applicability of the wild cwf approach to a prominent higher model in type theory.