It is a lot like Clinton impeachment were Clinton did everything he could to stop any and all investigations against him. in the end the conservatives got hurt.
In one sense of the word it is impossible. Probably not to prove it in a technical, evidential sense. But we must realize that if JC himself stood before us and said Hussein was ineligible, 40% of the people would not believe it. And a number of others would not care anyway.
So, what it comes down to is this. What will we do if it is actually proven he is not legally President and our governmental legal system refuses to depose him?