Gaunt-closure of the ∞-categorical Gray tensor product

Establish that for any pair of gaunt ω-categories A and B, the ∞-categorical Gray tensor product A ⊗ B in Cat_(∞,∞) is again gaunt; equivalently, show that the ∞-categorical and strict Gray tensor products coincide on the full subcategory Gaunt ⊂ Cat_(∞,∞) (or Gaunt ⊂ StrCat_ω).

Background

The paper develops Campion’s ∞-categorical Gray tensor product ⊗ on (∞,∞)-categories and compares it with the strict Gray tensor product ⊗^str on strict ω-categories. The authors work extensively with the full subcategory Gaunt of strict ω-categories that are simultaneously univalent and strict, and many constructions are phrased via pasting diagrams in Gaunt.

A key technical assumption that would simplify computations is the agreement of the ∞-categorical tensor product with the strict tensor product on Gaunt. This agreement is equivalent to the closure of Gaunt under ⊗ in the ∞-categorical setting; proving it would guarantee that tensor products of gaunt ω-categories remain gaunt and legitimize strictly diagrammatic calculations for objects like θ_p ⊗ θ_q. The authors note that this equivalence is expected but not yet proven.