Is every coNP-complete language P-isomorphic to an P-immune coNP-complete language? OR Is there a P-immune coNP-complete language? A set is $\mathsf{P}$-immune iff 
it has no non-trivial $\mathsf{P}$ subset.


  Is every $\mathsf{coNP}$-complete language $\mathsf{P}$-isomorphic to
  an $\mathsf{P}$-immune $\mathsf{coNP}$-complete language?


Joshua Grochow's answer to my previous questions shows the answer is negative assuming cryptographic conjectures.
Is it possible to show the answer is negative 
only assuming  $\mathsf{P}\neq\mathsf{NP}$?

Or show the answer as positive with  additional assumptions.

As @Kaveh points out based upon a P isomorphism assumption (Bertman-Hartmanis conjecture) one only has to show the existence of one such 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).