Recent work has argued that different truth-value gaps, such as homogeneity and implicatures, are in fact presuppositions, based on similarities in their pragmatic status and their projection properties (Fox, 2018; Bassi et al., 2021 a.o.). Nevertheless, these inferences seem to differ from presuppositions in some ways: (i) as evidenced by the Hey, wait a minute! test, homogeneity and implicatures don’t interact with the common ground in the same way as presuppositions (Spector, 2013; Bassi et al., 2021) and (ii) homogeneity and implicatures project differently from presuppositions in some environments, like the scope of certain quantifiers (Spector, 2013; Križ, 2015; Chatain and Schlenker, 2023). In this talk, I argue that under the view that homogeneity and implicatures are due to a syntactically represented presuppositional exhaustification operator pex, (i) follows from an independently needed redundancy constraint which requires that presuppositions due to pex are obligatorily accommodated with the A operator. I then show that on a modular view of presupposition projection, this result can be exploited to account for at least some of the differences in projection.