In Müger's article "Conformal Field Theory and Doplicher-Roberts Reconstruction", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore every premodular category, which is a special case, see Bruguières articles) is the subcategory of a modular category, namely of its closure.

Is it known when this inclusion is full? If not, is it possible for an arbitrary premodular category to construct a modular category that contains it as a full subcategory?