CFDLs wouldn't seem possible in 6,000 K hydrogen, due to its excellent conductivity. But there are two known forces that can keep electric charges separate in the absence of electrical resistance. They are (obviously) the two other forces present at the macroscopic level: the magnetic force, and gravity. We already ruled out magnetism, so we'll investigate the effects of gravity.
What about heat? Shouldn't you mention that here? Doesn't high heat keep charges separated?*18721 Is the temperature too low to do that at the surface? I think you said the surface is 4,600K, while the bottom of the photosphere is 6,400K. Is 4,600K hot enough to keep H and He etc ionized?