In the paper Sato-Wakui "[COMPUTATIONS OF TURAEV-VIRO-OCNEANU INVARIANTS OF 3-MANIFOLDS FROM SUBFACTORS](http://arxiv.org/abs/math/0208242)" 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?



Does the Turaev-Viro theory for the generalized $E_6$ subfactor for $\mathbb{Z}/7$ distinguish $L(7,1)$ and $L(7,2)$?
I think [this](http://www2.itc.kansai-u.ac.jp/~wakui/ILDT07wa.pdf) 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 $L(7,2)$.


