Suppose $coNP \neq NP$
language B would be called "complete" in $coNP-NP$ if:
- $B\in coNP - NP$
- $A\in coNP-NP \implies A\leq_pB$
Are there any "complete" languages in $coNP - NP$?
If we are assuming that $coNP≠𝑁𝑃$,
we can conclude that every language that is $co NP$ complete is not in $NP$ (a contradiction to your given assumption).
Thus, every language we already know of that is $coNP$ complete, is complete as well in $coNP -NP$.