Cyclotomic defining-field conjecture for modular categories

A modular category is a braided fusion category over $\mathbb C$ equipped with a compatible ribbon structure whose braiding is nondegenerate; a defining number field is a number field over which the category's structural data are defined. A field is cyclotomic if it is contained in a cyclotomic extension of $\mathbb Q$. Cyclotomic defining-field conjecture. Every modular category defined over $\mathbb C$ has a cyclotomic defining number field. This is motivated by the cyclotomicity of the $S$- and $T$-matrix data in known constructions and by the fact that Drinfeld centers of known exotic fusion categories have cyclotomic defining fields; a general proof is not known.