Epireflections and supercompact cardinals

We prove that the existence of arbitrarily large supercompact cardinals implies that every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if LL is a localization functor on a balanced accessible category such that the unit morphism X→LXX→LX is an epimorphism for all XX and the class of LL-local objects is defined by an absolute formula, then the existence of a sufficiently large supercompact cardinal implies that LL is a localization with respect to some set of morphisms.