Removing the multiplicatively strong hypothesis from the strictification result

Determine whether the up-to-adjunction strictification of symmetric bimonoidal functors between bipermutative categories can be achieved without assuming the functors are multiplicatively strong; equivalently, ascertain whether the strictification construction and associated bimonoidal adjunction extend to arbitrary symmetric bimonoidal functors whose multiplicative unit and monoidal constraints need not be isomorphisms.

Background

The main theorem of the paper establishes a strictification (up to adjunction) for symmetric bimonoidal functors under the assumption that they are multiplicatively strong, i.e., the multiplicative monoidal and unit constraints are isomorphisms. This assumption is used critically (e.g., to invert multiplicative constraints when defining the strictified functor on morphisms).

The authors explicitly note that it is unknown whether the multiplicatively strong hypothesis can be removed, leaving open the possibility of a stronger strictification theorem that applies to all symmetric bimonoidal functors between bipermutative categories.