Weak equivalence of the injective nerve and the ordinary nerve

Determine for which finite categories $C$ the inclusion of the injective nerve $N^{\iota}(C)\to N(C)$ into the ordinary nerve is a weak equivalence of simplicial sets.

Background

For a finite category $C$, the injective nerve $N^{\iota}(C)$ is formed from functors from finite ordinal categories that are injective on objects, whereas the ordinary nerve $N(C)$ includes all such functors and their degeneracies. The inclusion is always an embedding of simplicial sets, but the paper gives examples where the two nerves have different homotopy types, such as reachability categories associated with strongly connected directed graphs.

The paper leaves unresolved the precise categorical conditions under which omitting simplices with repeated objects preserves weak homotopy type. This question is motivated by the relationship between the injective nerve, directed flag complexes, and the regular magnitude-path spectral sequence.