Bestiary hypothesis for higher categories of vector spaces

Let $d\geq 0$. Consider the homotopy actions of the relevant orthogonal groups on the indicated higher categories of vector spaces. Framing-action triviality conjecture. The homotopy $\mathrm{O}(d+1)$-action on $\Sigma^d\mathbf{Vec}_\mathbb R$, the homotopy $\mathrm{SO}(d+1)$-action on $\Sigma^d\mathbf{Vec}_\mathbb C$, and the homotopy $\mathrm{Spin}(d+1)$-action on $\Sigma^d\mathbf{SVec}_\mathbb C$ are canonically trivializable. These claims would explain why the corresponding real, complex bosonic, and complex fermionic topological orders naturally define unoriented, oriented, and spin TQFTs, respectively. The source presents them as expectations based on examples, with no resolution supplied.