The gaunt-colimit characterization of flagged higher categories

Let $\operatorname{{\sf Cat}}_n$ denote the category of flagged higher categories, let $\operatorname{{\sf fCat}}_n$ be the category of flagged higher categories, and let $\operatorname{PShv}(\operatorname{{\sf Cat}}_n)$ denote presheaves on $\operatorname{{\sf Cat}}_n$. A gaunt colimit diagram is a functor $\mathcal J^{\operatorname{\triangleright}}\to\operatorname{{\sf Cat}}_n$ such that, for every $0\leq i\leq n$, its composite with $\operatorname{{\sf Cat}}_n\to\operatorname{{\sf Cat}}_i$ is a colimit diagram. The restricted Yoneda functor is fully faithful, with image consisting precisely of those presheaves that carry the opposites of gaunt colimit diagrams to limit diagrams.

Gaunt-colimit characterization. The restricted Yoneda functor

$$\operatorname{{\sf fCat}}_n\longrightarrow\operatorname{PShv}(\operatorname{{\sf Cat}}_n)$$

is fully faithful, and its image consists of exactly those presheaves that carry the opposites of gaunt colimit diagrams to limit diagrams.

This conjecture proposes an intrinsic characterization of flagged higher categories among presheaves on $\operatorname{{\sf Cat}}_n$, with gaunt colimits encoding colimits that do not generate invertible morphisms. The supplied text does not state whether the conjecture is known or remains open.