Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My understanding is that it is only the Morita class of the algebra which is important for determining the conformal field theory. Also, in the more general case of a fusion category, it is known by a result of Ostrik that up to Morita equivalence, separable algebras correspond to module categories for the fusion category.

Over ordinary vector spaces, every separable algebra is Morita equivalent to a commutative one, which of course makes the Morita equivalence classes easier to study. So this motivates my question: is every separable algebra in a modular tensor category Morita equivalent to a commutative one? If not, what about restricting to symmetric separable algebras?

I am working always over $\mathbb{C}$ here, and assuming that the categories involved are semisimple and with a finite number of isomorphism classes of simple objects.