On the structure of Witt groups and minimal extension conjecture

Let $\mathcal{E}=\text{Rep}(G)$ be a Tannakian fusion category. For a braided fusion category $\mathcal{C}$ over $\mathcal{E}$ we give sufficient and necessary conditions that characterize the Witt relation $[\mathcal{C}]=[\mathcal{E}]$. Then we show the Witt group $\mathcal{W}(\mathcal{E})$ is naturally a direct sum of Witt group $\mathcal{W}:=\mathcal{W}(\text{Vec})$ and the group ${\text{H}}^4(G,{\mathbb{K}}^{\times })$. Consequently, for any non-degenerate fusion category $\mathcal{C}$ over $\mathcal{E}$, there is a positive integer $n$ (e.g. $n=\mathrm{\mid }G\mathrm{\mid }$) such that ${\mathcal{C}}^{{⊠}_{\mathcal{E}}^n}$ admits a minimal extension.