On a conjecture of Kac-Wakimoto

We prove a conjecture about minimal index of certain representations of Coset Algebraic Conformal Field Theories under certain conditions as formulated previously by us. As a by-product, the Kac-Wakimoto Conjecture (KWC) which is related to the asympotics of the coset characters is true under the same conditions. The same idea in the proof also proves a recent conjecture related to subfactors from conformal inclusions.