We use antecedent-final conditionals to formulate a challenge to parsing-based theories of local contexts, and associated theories of presupposition projection and triviality, like the one given in Schlenker 2009. We show that a theory like Schlenker’s predicts that the local context for the antecedent of an antecedent-final conditional will entail the negation of the conditional’s consequent. It thus predicts that presuppositions triggered in the antecedent of antecedent-final conditionals will be filtered if the negation of the consequent entails the pre-supposition. But this is wrong: John isn’t in Paris, if he regrets being in France intuitively presupposes that John is in France, contrary to this prediction. Likewise, parsing-based approaches to triviality predict that material entailed by the negation of the consequent will be felt to be redundant in the antecedent of the conditional. But this is wrong: John isn’t in Paris, if he’s in France and Mary is with him is intuitively felicitous, contrary to this prediction. Importantly, given that the material in question appears in sentence-final position in antecedent-final con-ditionals, both incremental (left-to-right) and symmetric versions of parsing-based theories of local contexts make the same problematic predictions here. In Mandelkern and Romoli 2017, we discuss one solution to this problem, given within a broadly parsing-based pragmatic approach. In this paper, we explore an alternate direction: incorporating attention to hierarchical structure into the calculation of local contexts. We sketch several possible implementations and point to some of the possibilities and challenges for a hierarchical approach to local contexts.