Isometric enhancement of the equivalence Fun^{\dag,\vee}(B\cC \to Mod^\dag(\cD)) \cong \sH^*mFC(\cC \to \cD)

Show that the equivalence of 2-categories Fun^{\dag,\vee}(\mathrm{B}\cC \to Mod^\dag(\cD)) \cong \sH^*mFC(\cC \to \cD) is an isometric equivalence of 3-Hilbert spaces, i.e., that it preserves the 3-Hilbert space structures (unitary adjoint functors and spherical weights) exactly, not just up to unitary equivalence.

Background

In establishing a unitary 3-equivalence between the 3-category of H*-multifusion categories and the 3-category of 3-Hilbert spaces, the authors identify a natural equivalence between the hom 2-categories of UAF-preserving functors out of B\cC and the bimodule morphisms \sH^*mFC(\cC \to \cD).

While the underlying 2-categorical equivalence is proved, the authors conjecture that it refines to an isometric equivalence at the level of 3-Hilbert spaces, meaning that the associated weights and adjoint structures are strictly preserved. Confirming this would tighten the correspondence between functorial and bimodule-theoretic descriptions within the 3-Hilbert framework.