Categorical hypercompleteness conjecture for finite and infinite categories

Let $n\text{-}\operatorname{Cat}$ denote the category of $n$-categories, let $\operatorname{Surj}_{\omega}$ denote the class of $\omega$-surjective morphisms, and let $\omega\text{-}\operatorname{Cat}^{\mathrm{coind}}$ be the full subcategory of $\omega\text{-}\operatorname{Cat}$ consisting of objects local with respect to $\Sigma_nE_{eq}^{\mathrm{coind}}\to \mathbf{D}_n$ for every $n\in\mathbb{N}$, so that their coinductive $\omega$-equivalences are trivial. Categorical hypercompleteness conjecture. For every $n\in\mathbb{N}$, the morphism

$$n\text{-}\operatorname{Cat}\to n\text{-}\operatorname{Cat}[\operatorname{Surj}_{\omega}^{-1}]$$

is an equivalence. Equivalently, every $\omega$-surjective morphism between $n$-categories is an equivalence. Moreover, the morphism

$$\omega\text{-}\operatorname{Cat}^{\mathrm{coind}}\to n\text{-}\operatorname{Cat}[\operatorname{Surj}_{\omega}]$$

is an equivalence. In other words, the categorical hypercompletion of $\omega\text{-}\operatorname{Cat}$ consists of $\omega\text{-}\operatorname{Cat}^{\mathrm{coind}}$. The assertion concerns whether iterated surjectivity detects equivalences in finite dimensions and identifies the corresponding hypercompletion in the infinite-dimensional setting; its status is not established in the supplied text.