The Turaev-Viro-Barrett-Westbury invariant of a closed oriented topological $3$-manifold $M$ for a spherical fusion category $\mathcal{C}$ is a number denoted $|M|_{\mathcal{C}}$ computed from (but independent of the choice of) a triangulation of $M$ and $\mathcal{C}$-data labelings.

See the book Turaev-Virelizier (2017) on Section 13.1, and the papers Turaev-Viro (1991) and Barrett-Westbury (1996).

**Question**: Are there non-homeomorphic closed oriented $3$-manifolds $M$ and $N$ such that $|M|_{\mathcal{C}} = |N|_{\mathcal{C}}$, for all spherical fusion category $\mathcal{C}$?

This answer by Meng Cheng provides a formula for $|M|_{\mathcal{C}}$ with $M=L(p,q)$ a Lens space and $\mathcal{C}=Vec_{G,\omega}$ a pointed fusion category, and shows that it distinguishes $L(7,1)$ from $L(7,2)$, for a good choice of $G$ and $\omega$.