Isometric nature of the relative Deligne product associativity constraint

Prove that the canonical unitary associativity constraint between the relative Deligne products M ⊠_C (N ⊠_D O) and (M ⊠_C N) ⊠_D O of H*-multifusion module categories is an isometric equivalence with respect to the module trace structures constructed for these relative Deligne products.

Background

In the paper the authors define relative Deligne products of module and bimodule categories over H*-multifusion categories and equip these with module traces, yielding unitary structures. They then observe that both parenthesizations M ⊠_C (N ⊠_D O) and (M ⊠_C N) ⊠_D O represent the same balanced-functor universal property, providing a canonical unitary associativity constraint between them.

While the existence of this unitary constraint is established, the authors conjecture that it is also isometric, i.e., that it preserves the relevant trace data (and hence norms) exactly, not merely up to unitary equivalence. Establishing this would strengthen the coherence of the relative Deligne product in the isometric sense within the framework of 3-Hilbert spaces.