The commuting gauging diagram for fermionic symmetry-enriched string-net models

Let $G$ be a symmetry group, let $\mathcal{S}_G$ be a $G$-graded super fusion category with trivial $\omega_2$, and let $\mathcal{S}_1$ be its trivial grading sector. The category $\mathcal{Z}_1(\mathcal{S}_1)$ carries a $G^f$-action, where $G^f$ includes fermion parity. A $G$-crossed extension is the extension used to gauge $G$, followed by $G$-equivariantization. The commuting-gauging conjecture. After gauging $G$, the resulting category is

$$\mathcal{Z}_1({}^{\mathrm{sVect}}\mathcal{C}_G)\cong{}^{\mathrm{sVect}}\mathcal{Z}_1(\mathcal{C}_G)_0,$$

and the diagram comparing $G$-gauging, fermion-parity gauging, and $G^f$-gauging commutes. The conjecture asserts compatibility between the fermionic and bosonic gauging procedures; the source notes that it uses a $G$-crossed extension on the super modular category to make this diagram symmetric.