The characterization of 3-Hilbert spaces by inductive fixed-point data

Let $\fX\in 3\mathsf{Vect}$ be an $O(3)$-fixed point. For every $F\colon 2\mathsf{Vect}\to\fX$, suppose that $F$ is equipped with an $O(2)$-fixed point structure, and that $F^{\dagger_b}F$ has a lift to a $2$-Hilbert space. 3-Hilbert space conjecture. An $O(3)$-fixed point $\fX\in 3\mathsf{Vect}$ equipped with this data exactly corresponds to a $3$-Hilbert space as defined in the cited work. This conjecture asserts that the inductive fixed-point construction gives precisely the proposed notion of a $3$-Hilbert space.