In order to handle presuppositions in the scope of attitude verbs, the binding theory allows presuppositions triggered in a subject's beliefs to be bound at the matrix level; and it allows presuppositions triggered in non-doxastic attitudes to be bound in the subject's beliefs (Geurts, 1999; Maier, 2015). However, we argue that this leads to serious overgeneration, for example it predicts that the unacceptable `Sue will come to the party, but Bill is sure that she won't and that only Sue will (...) come to the party' should be equivalent to the acceptable `Sue will come to the party, but Bill is sure that nobody will come to the party'. This is because the presupposition triggered by `only Sue will come to the party' should be able to be bound at the matrix level. We discuss some responses to this problem, but argue that they all have shortcomings. (shrink)