The wording there makes it rather obvious that the Constitution outrankes the treaty -- but even a moment's thought on the nature of authority would reveal that it cannot be otherwise: the one who sends/authorizes/commissions is of a greater authority than the one sent/authorized/commissioned (SAC). If the SAC acts outside the constraints that they were bound by, then they are not acting legitimately, likewise the authorizer cannot grant unto the SAC greater authority/power than the authorizer has. Therefore, because the Constitution places limits on the government, the government cannot validly bind the people to a treaty that is contrary to the Constitution.
I agree.