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

A detailed discussion of your question can be found in Davidovitch et. al's paper "On arithmetic modular tensor categories". 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.