Stricter discrete opfibrations in multicategories

Determine whether there exist pairs of diagrams in a multicategory that are weakly equivalent when weak equivalence is defined using the stricter notion of discrete opfibration that allows lifts against multimorphisms with a lifted domain, and, if so, provide explicit examples or characterize such cases.

Background

In the general 2-categorical setting, the paper adopts a broad notion of discrete opfibration (Definition 2-categorical-dopf) that suffices to prove the main localization results for multicategories.

The authors note that, specifically for multicategories, there exists a natural stricter notion of discrete opfibration that permits lifts against multimorphisms with a lifted domain. While the main results are established for the broader notion, the status of weak equivalences defined using the stricter notion is left unresolved.

The open question concerns whether there are concrete, nontrivial examples of diagrams in multicategories that become weakly equivalent only when the stricter lifting requirement is used, potentially revealing a difference between the broader and stricter notions of discrete opfibration in practice.