The interactions between scalar implicatures and presuppositions have received more and more attention in the recent literature. One puzzle of primary interest comes from the observation that implicatures can arise in the presupposi- tional component of a sentence without also arising in its assertion part (Sharvit & Gajewski 2008, Magri 2009, Gajewski & Sharvit 2012 a.o.). Some researchers have taken this observation as an argument that the mechanism deriving implica- tures should be extended to the presuppositional level. Spector & Sudo (2017) have recently challenged this view on different grounds and proposed an alternative account on which implicatures targeting presuppositional contents arise through a dedicated form of scalar strengthening. In this paper, we present a novel data point, which provides a new perspective on the interaction between implicatures and presuppositions: we observe that a sentence like Noah didn’t realise that Olivia can take Logic or Algebra has a reading on which a free choice inference appears at the presuppositional level (i.e., it suggests that Olivia can choose between the two classes), but not at the assertion level (i.e., it conveys that Noah doesn’t be- lieve that she can take either one). As we discuss, these cases are challenging forSpector & Sudo’s (2017) account for two reasons: it does not derive the intended reading and incorrectly predicts such sentences to be odd in contexts where the free choice inference is common knowledge. We discuss two possible ways for addressing these challenges, as well as those previously raised by Spector & Sudo 2017. The first one retains the system in Spector & Sudo 2017 and combines it with a homogeneity-based account of free choice (Goldstein 2018). The second one retains the implicature-based approach to free choice (Fox 2007, Bar-Lev & Fox 2017 a.o.), and integrates it with a theory of implicatures where meaning strengthening operates similarly – yet in some way independently – at the assertion and presuppositional levels, as suggested in Marty (2017).