The generalized E6-subfactor conjecture on distinguishing lens spaces

Let $L(7,1)$ and $L(7,2)$ be the indicated lens spaces, and suppose there exists a generalized $E_6$-subfactor with group symmetry $G=\mathbb{Z}/7\mathbb{Z}$. Let the Turaev–Viro–Ocneanu invariant from this subfactor be the associated invariant of a 3-manifold. Generalized $E_6$-subfactor conjecture. The invariant distinguishes $L(7,1)$ and $L(7,2)$; equivalently, their associated invariant values are different. The conjecture is motivated by the preceding computations for generalized $E_6$-subfactors with other group symmetries, which give equal values for the pairs tested, and predicts that the as-yet hypothetical $G=\mathbb{Z}/7\mathbb{Z}$ case detects these lens spaces.