Mueger showed in this paper that if $C$ is a modular fusion category and $D$ is a modular fusion subcategory of $C$, then $C$ is equivalent to $D \boxtimes M_C(D)$, where $M_C(D)$ is the Mueger centralizer of $D$ in $C$ (which is incidentally also modular).

Does this factorization of $C$ somehow impart a factorization of e.g. the Reshetikhin-Turaev TFT $\mathcal{Z}^{RT}_{C}$ built from $C$? Or maybe the other way around, is knowing the TFTs $\mathcal{Z}^{RT}_{D}$ and $\mathcal{Z}^{RT}_{M_C(D)}$ enough to (somehow) obtain $\mathcal{Z}^{RT}_C$?

Mueger showed in this paper that if $C$ is a modular fusion category and $D$ is a modular fusion subcategory of $C$, then $C$ is equivalent to $D \boxtimes M_C(D)$ as ribbon categories, where $M_C(D)$ is the Mueger centralizer of $D$ in $C$ (which is incidentally also modular).

Does this factorization of $C$ somehow impart a factorization of e.g. the Reshetikhin-Turaev TFT $\mathcal{Z}^{RT}_{C}$ built from $C$? Or maybe the other way around, is knowing the TFTs $\mathcal{Z}^{RT}_{D}$ and $\mathcal{Z}^{RT}_{M_C(D)}$ enough to (somehow) obtain $\mathcal{Z}^{RT}_C$?