Yes, exactly. Unless and until Congress discovers Obama's infirmity and he is legally dismissed from office, he's the sitting President of the United States.
Would a civilian court hearing the case of a road crew manager who refused to plow a street upon the Mayor’s order allow the refusenik the ability to contest the Mayor’s eligibility for office, to demand proof of birth place, citizenship and age aided by power of subpoena, oath and subject to cross-examination?