Inspired by Suresh's post, for a new problem in $\mathsf{coNP}$, whose true proof complexity is intermediate between $\mathsf{NP \cap coNP}$ and being coNP-complete, I am interested in methods which show that it is probably hard to prove the existence of good characterization.What techniques and methods do exist to show that a $coNP$ problem might not have good characterization?
Techniques for showing intermediate status between $\mathsf{NP \cap coNP}$ and coNP-completeness
cc.complexity theory;proof complexity