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In the paper Sato-Wakui "COMPUTATIONS OF TURAEV-VIRO-OCNEANU INVARIANTS OF 3-MANIFOLDS FROM SUBFACTORS" they compute certain Turaev-Viro-Ocneanu invariants of certain lens spaces. One of the results is that the generalized $E_6$ subfactor for $\mathbb{Z}/p$ distinguishes the lens spaces $L(p,1)$ and $L(p,2)$ for $p=3,5$, and they conjecture (or at least raise the question) of whether this holds for $p=7$.

I am wondering if any progress has been made on this question since this paper appeared?

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I think this paper of Wakui says that the answer is "No". The Turaev-Viro invariants associated to the generalized $E6$ subfactors for $\mathbb{Z}/7$ don't seem to distinguish $L(7,1)$ and $L(7,2)$.

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