Strictness of Mnd ⊗ Mnd and Adj ⊗ Adj

Determine whether the ∞-categorical Gray tensor products Mnd ⊗ Mnd and Adj ⊗ Adj are strict (i.e., gaunt ω-categories), equivalently whether they agree with the strict Gray tensor products Mnd ⊗^str Mnd and Adj ⊗^str Adj.

Background

The construction of the universal shear map and its analysis in the paper leverages strict orientals because the authors cannot assume that the ∞-categorical tensor products Mnd ⊗ Mnd and Adj ⊗ Adj are already strict (gaunt). If strictness were known, certain arguments could proceed directly in the strict ω-categorical setting without detours.

Resolving strictness would also clarify the relationship between the ∞-categorical Gray tensor product and its strict counterpart in these key examples (the walking monad and walking adjunction), which underpin the identification of bialgebras and the invertibility of universal shear maps.