MZ operation for moderate-growth pre-Tannakian categories

Prove that for every finitely tensor-generated pre-Tannakian category of moderate growth over the base field, the MZ operation yields the representation category of a finite group; specifically, establish that MZ(C) is equivalent to Rep(K) for some finite group K and, consequently, that applying the MZ operation twice gives (MZ)^2(C) ≅ Vec_k.

Background

The paper studies the MZ operation, defined as taking the Müger center of the Drinfeld center, and analyzes its behavior across numerous classes of tensor categories. For finite tensor categories Z(C) is known to be non-degenerate, and for several families (e.g., connected affine group schemes, Frobenius exact categories) the authors identify strong constraints on MZ(C).

Motivated by these results and examples, they formulate a conjecture predicting that for any finitely tensor-generated pre-Tannakian category of moderate growth, MZ(C) must be the representation category of a finite group, so two iterations of MZ collapse to Vec_k.

References

This motivates the following conjecture. If $C$ is a finitely tensor-generated pre-Tannakian category of moderate growth, then $MZ(C) = \Rep(K)$, where $K$ is a finite group. Thus $(MZ)2(C) = Vec_k$.

— The Drinfeld center of an oligomorphic tensor category  (2604.00290 - Etingof et al., 31 Mar 2026) in Section 9.2 (The MZ operation), Conjecture