Classification of unitary modular tensor categories (UMTCs) Context/background:
I'm approaching this topic from the perspective of anyonic systems.
In the study of anyons, one works with fusion categories. Of course, for physicality, we demand that
i) The category is unitary (all $F$ and $R$ symbols are unitary) i.e. a UTC. 
ii) The category is modular ($S$ matrix is unitary - this ensures nondegenerate braiding) i.e. an MTC. 
Therefore, we are interested in classifying all UMTCs (which may play host to a physical theory of anyons in Nature). The advantage of working with MTCs is that demanding $S$ unitary provides a nice mechanism for discarding fusion rules that are unphysical (a technique that is used by Wang et al. in https://arxiv.org/abs/0712.1377).
Questions:
1) Given modular fusion rules $\mathcal{N}$, how do we count the number of corresponding UMTCs?
The obvious way seems to be to count the no. of distinct unitary solutions of $F$ and $R$ symbols (upto gauge freedom). However, it is conjectured that the $S$ and $T$ matrices uniquely define a UMTC, and so we might count the number of 'modular symbols' $(\mathcal{N}; S, T)$ corresponding to $\mathcal{N}$ - only this increases the count due to symmetries $(\mathcal{N}; -S, T)$, $(\mathcal{N}; S^{\dagger}, T^{\dagger})$ of the symbol. Which way is the more appropriate method for counting UMTCs w.r.t $\mathcal{N}$? The latter method seems stronger yet based on conjecture (though I suppose this is harmless in practice).
2) Are there any known examples of modular $\mathcal{N}$ with no unitary solutions for $F$ and $R$ symbols?
3) Are there examples of fusion rules $\mathcal{N}$ that yield UTCs which aren't modular?
 A: *

*Recently, Mignard and Schaunberg found a counter example to the conjecture (https://arxiv.org/abs/1708.02796). Counting $(S,T)$ pairs is still just a shorthand way of counting monoidal equivalence classes, which is not exactly the same as counting gauge classes (it's close though). Taking a note along that line though, it is known that for whatever fusion rules you want to study there exist polynomial invariants (https://arxiv.org/abs/1509.03275) which will get the job done at all levels from fusion to modular.

*Eric Rowell showed (https://arxiv.org/abs/math/0503226) that the braided tensor categories coming from $\mathcal C(\mathfrak{so}_{2r+1},l,q)$ for $l$, $2(2r+1)<l$, and $q^2$ a primitive $l$-th root of unity are not unitarizable (this implies no unitary solutions to pentagon and hexagon equations).

*The categories above are modular if and only if $r$ is odd (see theorems 4.3 and 4.5 of Rowell).
This answer is related: Is the representation category of quantum groups at root of unity visibly unitary?
