Grothendieck's homotopy hypothesis for weak globular infinity-groupoids (Ara-Henry coherators)

Theorem 1.1 (Henry's pushout conjecture): for every Grothendieck coherator in the Ara-Henry convention, elementary expansions of cellular models are weak equivalences. Section 5 derives the canonical left semi-model structure and, through Henry's comparison, a Quillen equivalence with spaces, so the homotopy theory is independent of the coherator within that convention. It does not cover coherators outside the Ara-Henry convention, Grothendieck's weak $\infty$-categories (the non-invertible case), or Taylor's generalized pushout conjecture for algebraic coherators.

Theorem 1.1 (Henry's pushout conjecture): for every Grothendieck coherator in the Ara-Henry convention, elementary expansions of cellular models are weak equivalences. Section 5 derives the canonical left semi-model structure and, through Henry's comparison, a Quillen equivalence with spaces, so the homotopy theory is independent of the coherator within that convention. It does not cover coherators outside the Ara-Henry convention, Grothendieck's weak $\infty$-categories (the non-invertible case), or Taylor's generalized pushout conjecture for algebraic coherators.