Reasons to believe $P \ne NP \cap coNP$ (or not) It seems that many people believe that $P \ne NP \cap coNP$, in part because they believe that factoring is not polytime solvable. (Shiva Kintali has listed a few other candidate problems here). 

On the other hand, Grötschel, Lovász, and Schrijver have written that "many people believe that $P=NP\cap coNP$." This quote can be found in Geometric Algorithms and Combinatorial Optimization, and Schrijver makes similar statements in Combinatorial Optimization: Polyhedra and Efficiency. This picture makes it clear where Jack Edmonds stands on the issue.

What evidence supports a belief in $P\ne NP\cap coNP$? Or to support $P=NP\cap coNP$?
Theorem 3.1 of One-Way Permutations and Self-Witnessing Languages C. Homan and M. Thakur, Journal of Computer and System Sciences, 67(3):608-622, November 2003. [ as .pdf ] states that $P≠UP∩coUP$ if and only if ("worst-case") one-way permutations exist. Theorem 3.2 recalls 10 further hypotheses that have been shown to be equivalent to $P≠UP∩coUP$. 

Also, we have strong reason to conjecture that $UP \ne NP$. Therefore, the above theorem and the conjecture result in a strong reason to believe that $P \ne NP ∩ coNP$. 



Disclaimer: I moved Mohammad Al-Turkistany's edit of my answer to this community wiki answer. He believes that it perfectly answers the question since the existence of one-way permutations is widely believed. I myself haven't yet sufficiently understood the difference between "worst-case" and "average-case" one-way functions to claim that it really answers the question.