A dynamic analysis of modalized questions faces several difficulties. When an existential modality scopes over the wh-phrase, two kinds of phenomena are observed. First, the external staticity of the modal operator prevents the wh-phrase from being referred back to in indicative follow-up sentences. Anaphoric reference to the wh-phrase should yet be allowed through modal subordination. Second, exhaustivity and global uniqueness presupposition are weakened in modalized which questions, leading to mention-some readings and local uniqueness presupposition. We design a model capturing these two desiderata, based on dynamic inquisitive semantics and stack-based semantics for local contexts. Our model provides a uniform analysis of modalized and conditional questions using the Kratzerian theory of modality. It also correctly predicts discrepancies in presupposition projection between modalized and conditional questions.