A detailed discussion of your question can be found in Davidovitch et. al's paper "[On arithmetic modular tensor categories][1]". They say that it is still an open problem whether there are non-cyclotomic modular tensor categories. However, they do conjecture formally (conjecture 4.5) that every MTC is cyclotomic. I am not aware of any significant recent progress on the conjecture.


  [1]: https://arxiv.org/pdf/1305.2229.pdf