Is every coNP-complete language P-isomorphic to an P-immune coNP-complete language? OR Is there a P-immune coNP-complete language?
Josh's answer uses two conjectures and 
both of them are considered to be highly likely to be correct by experts 
even if not proven yet.
A positive answer means that at least one of the two conjecture is incorrect.
That would be a major very surprising result.
In other words, 
it is highly unlikely that the answer to your question is positive, and
even if the answer is positive, 
it is not possible to prove it at this time 
(otherwise we would have already refuted one of those conjectures).