Classification conjecture for minimal modular extensions of 2Rep(G)

Let $G$ be a finite group. A minimal modular extension of $\operatorname{2Rep}(G)$ is a unitary modular tensor 2-category equipped with a braided monoidal fully faithful embedding $\operatorname{2Rep}(G)\hookrightarrow\mathcal{C}$ such that the relative sylleptic center of $\operatorname{2Rep}(G)$ in $\mathcal{C}$ is braided monoidally equivalent to $\operatorname{2Rep}(G)$. Classification conjecture. The equivalence classes of minimal modular extensions of $\operatorname{2Rep}(G)$ are classified by

$$H^4(G,\mathbf{k}^{\times}).$$

This is motivated by the classification of symmetry-protected topological orders in lower and higher dimensions. The paper establishes that $\mathcal{Z}(\operatorname{2Vec}_G^\omega)$ gives a minimal modular extension for each $\omega\in Z^4(G,\mathbf{k}^{\times})$, but the claimed classification of all such extensions remains open.