The black dwarf would endure till the end of the universe.
The white dwarf begins hot and white and ends as a black dwarf cold and dark after many billions of years according to theory. if we assume that a white dwarf becomes a black dwarf at say 5K then itll take at least 10^15 years for one to form and this is the greater than the current age of the universe. So the concept itself being purely theoretical there cannot be any answer. All we can say is that black dwarfs would remain as one of the final objects when the universe ends.