Classical versions of Maximize Presupposition! (MP) posit a competition between contextually equivalent sentences that differ in regard of the logical strength of their presuppositions. Yet recent work has put forward novel MP-like cases which fall beyond the empirical scope of MP precisely because the relevant competitors are not contextually equivalent to their base sentence. To account for these cases,Spector & Sudo (2017) propose a novel principle, the Presupposed Ignorance Principle (PIP), whose formulation parallels that of MP but leaves out the former condition on contextual equivalence. This amendment of MP allows the PIP to capture the novel cases while preserving the classical ones. In this paper, we show that the PIP, however, overgenerates in a variety of other cases, and we argue that the culprit is precisely dropping the condition on contextual equivalence. We discuss two directions moving forward, based on recent proposals in the literature. The first is the Logical Integrity approach by Anvari 2019, 2018, which, among other things, replaces contextual equivalence with contextual entailment. As we discuss, this approach cannot account for our problematic cases either, and thus doesn’t fare much better than Spector & Sudo (2017) in this respect. The second is the implicature-based approach combining insights from Magri 2009, Marty 2017, 2019and Meyer 2013, which, among other things, reintroduces contextual equivalence. Just like MP, this approach accounts for our cases but it cannot capture, as it is, the novel cases by Spector & Sudo (2017). We sketch, however, how this approach can be extended to account for such cases as well. And, against the outlined empirical landscape, we take this to be the most promising strategy at this stage: retaining the contextual equivalence restriction from the original MP approach while accounting for the novel cases by Spector & Sudo (2017) in other ways.