Is the mechanism behind presupposition projection and filtering fundamentally asymmetric or symmetric? This is a foundational question for the theory of presup- position which has been at the centre of attention in the literature recently (Schlenker 2008b, 2009; Rothschild 2011/2015 a.o.). It also bears on broader issues concerning the source of asymmetries observed in natural language: are these simply rooted in superficial asymmetries of language use (since language use unfolds in time, which we experience as fundamentally asymmetric) or are they, at least in part, directly encoded in linguistic knowledge and representations? In this paper we aim to make progress on these questions by exploring presupposition projection across conjunction, which has typically been taken as a central piece of evidence that pre- supposition filtering is asymmetric in general. As a number of authors have recently pointed out, however, the evidence which has typically been used to support this conclusion is muddied by independent issues concerning judgments of redundancy, and additional concerns arise with regards to the possibility of local accommodation. We report on a series of experiments, building on previous work by Chemla & Schlenker (2012); Schwarz (2015), using inference and acceptability tasks, which aim to control for both of these potential confounds. In our results, we find strong ev- idence for asymmetric left-to-right filtering across conjunctions, but no evidence for right-to-left filtering—even when right-to-left filtering would, if available, rescue an otherwise unacceptable sentence. These results suggest that presupposition filtering across conjunction is indeed asymmetric, contra suggestions in the recent literature (Schlenker 2008a, 2009 a.o.) and paves the way for the investigation of further questions about the nature of this asymmetry and presupposition projection more generally. Our results also have broader methodological and theoretical implications: we find important differences in the verdicts of acceptability versus inference tasks in testing for projected content, which has both methodological ramifications for the question of how to distinguish presupposed content, and theoretical repercussions for understanding the nature of projection and presuppositions more generally.