Will a black hole eventually turn into a neutron star?

I think that once a black hole forms then that is it, because although its mass is finite, its density (in GR) becomes infinite at the central singularity. The loss of energy(mass) will then result in a shrinkage of the event horizon but no change in the black hole nature - the BH nature of an object is not determined solely by its mass, the density of the object is crucial.