Higher twisted Drinfeld center equivalence conjecture

Let $G$ be a finite group, let $\tau_h$ denote the transgression map, and let $\omega\in Z^{n+2}(G,\mathbf{k}^{\times})$. Suppose that a suitable notion of an $n$-category has been defined, and write $n\mathrm{Vec}_G^\omega$ for the corresponding twisted monoidal $n$-category. Higher center conjecture. There is an equivalence of $n$-categories

$$\mathcal{Z}(n\mathrm{Vec}_G^\omega) \simeq \boxplus_{[h] \in \operatorname{Cl}} \,\, n\mathrm{Rep}(C_G(h), \tau_h(\omega)).$$

This proposes the higher-categorical analogue of the twisted Drinfeld double description for 1-categories. The statement remains conditional on defining the relevant notion of an $n$-category and is presented as an expectation rather than an established result.