I'm now second-guessing an assertion I made here so let me ask it as a question.

Let $Cat_n$ be the 1-category of strict $n$-categories;

Let $\widetilde{Cat_n}$ be the $(\infty,1)$-category obtained from $Cat_n$ by localizing at the weak equivalences (as presented by the folk model structure);

Let $Gaunt_n \subseteq Cat_n$ be the fully faithful inclusion of the gaunt $n$-categories, i.e. those strict $n$-categories where every equivalence is an identity.

Let $\widetilde{Cat_n} \to Cat_{(\infty,n)}$ be the "inclusion" functor (which is probably

*not*fully faithful for $n \geq 3$).

**Questions:**

Is the composite functor $Gaunt_n \to Cat_n \to \widetilde{Cat_n}$ a fully faithful functor of $(\infty,1)$-categories?

Is the composite "inclusion" functor $Gaunt_n \to Cat_n \to \widetilde{Cat_n} \to Cat_{(\infty,n)}$ a fully faithful functor of $(\infty,1)$-categories?

I'm pretty sure the answer to (1) is *yes*, and intuitively the answer to (2) should also be *yes*, but I'm not quite sure.

Barwick and Schommer-Pries do show that if we further restrict to the inclusion $\Theta_n \to Cat_{(\infty,n)}$, or even the slightly larger $\Upsilon_n \to Cat_{(\infty,n)}$, then we get a fully faithful $(\infty,1)$-functor. But I'm not sure about all gaunt $n$-categories.