Equivariantization and delooping commute for multifusion categories

Let $G$ be a group. For a multifusion category $\mathcal{C}$ with a $G$-action, write $\mathcal{C}^G$ for its equivariantization and $\mathbf{Mod}(\mathcal{C})$ for its delooping. For a braided multifusion category $\mathcal{A}$ with a $G$-action, write $\mathcal{A}^G$ for its equivariantization. The resulting module 2-categories carry their natural structures over $\mathbf{2Rep}(G)$. Equivariantization–delooping conjecture. There should be equivalences of module 2-categories over $\mathbf{2Rep}(G)$ and of $\mathbf{2Rep}(G)$-module multifusion 2-categories, respectively:

$$\mathbf{Mod}(\mathcal{C})^G \simeq \mathbf{Mod}(\mathcal{C}^G),$$

and

$$\mathbf{Mod}(\mathcal{A})^G \simeq \mathbf{Mod}(\mathcal{A}^G).$$

The claim predicts that equivariantization commutes with delooping in both the multifusion and braided multifusion settings; the surrounding discussion presents the relevant equivariantization and de-equivariantization constructions for fusion 2-categories only as a rough sketch, indicating that the details remain to be established.