The trivialization conjecture for the orthogonal action on rigid higher categories

Let ${\mathsf{RigidCat}}_n$ denote the category of rigid symmetric monoidal $n$-categories, let $O_n$ act on it through the action supplied by the adjoint-category conjecture, and let $\iota_k\mathcal{C}$ be the $k$-category obtained from a rigid $n$-category $\mathcal{C}$ by forgetting higher non-invertible morphisms. The trivialization conjecture for the orthogonal action on rigid higher categories. The $O_n$-action on ${\mathsf{RigidCat}}_n$ canonically trivializes. It induces an $O_{n-k}$-action on $\iota_k\mathcal{C}$ for every rigid $n$-category, and this agrees with the $O_{n-k}$-action induced by the cobordism hypothesis. The conjecture is motivated by the expectation that a rigid symmetric monoidal category admits infinitely many deloopings with adjoints; its general validity remains open.