The finite-group conjecture for Müger centers of pre-Tannakian categories

Let $\mathcal{C}$ be a finitely tensor-generated pre-Tannakian category of moderate growth. Write $\mathcal{M}\mathcal{Z}(\mathcal{C})$ for its Müger center, and let $\mathrm{Vec}_k$ denote the category of finite-dimensional vector spaces over $k$. Finite-group conjecture. There is a finite group $K$ such that

$$\mathcal{M}\mathcal{Z}(\mathcal{C})=\operatorname{Rep}(K).$$

Consequently,

$$(\mathcal{M}\mathcal{Z})^2(\mathcal{C})=\mathrm{Vec}_k.$$

This conjecture generalizes the preceding result for finitely tensor-generated super-Tannakian categories, where the Müger center is the representation category of a finite quotient group. It predicts that the second iteration of the Müger-center operation is always the category of finite-dimensional vector spaces over $k$ in the stated moderate-growth setting.