Tannakian obstruction conjecture for braided fusion categories

Let $G$ be a finite group, let $\mathbf{Rep}(G)$ denote its representation category, and let $\mathcal B$ be a braided fusion category with obstruction class $O_4(\mathcal B)$. Write $\Obs(\mathbf{Rep}(G))$ for the obstruction group, $\Witt$ for the Witt group, and $\Witt(\mathbf{Rep}(G))$ for the Witt group relative to $\mathbf{Rep}(G)$. Then $O_4(\mathcal B)$ induces an isomorphism

$$\Obs(\mathbf{Rep}(G)) \cong \operatorname{coker}\bigl(\Witt \longrightarrow \Witt(\mathbf{Rep}(G))\bigr) \cong \operatorname{H}^4(\mathrm B G; \mathbbm{k}^\times).$$

Tannakian obstruction conjecture. The displayed isomorphism holds. This would complete the obstruction theory for $\mathbf{Rep}(G)$ by identifying every class in $\operatorname{H}^4(\mathrm B G;\mathbbm{k}^\times)$ with the obstruction of a braided fusion category. The text sketches a construction using a finite-group extension that trivializes the pulled-back class, but the conjecture is presented subject to the assumptions made there.