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Among many solutions for extracting useful answers from incomplete data, ontology-mediated queries (OMQs) use domain knowledge to infer missing facts. We propose an extension of OMQs that allows us to make certain assumptions—for example, about parts of the data that may be unavailable at query time, or costly to query—and retrieve conditional answers, that is, tuples that become certain query answers when the assumptions hold. We show that querying in this powerful formalism often has no higher worst-case complexity than in plain OMQs, and that these queries are first-order rewritable for DL-Liteℛ. Rewritability is preserved even if we allow some use of closed predicates to combine the (partial) closed- and open-world assumptions. This is remarkable, as closed predicates are a very useful extension of OMQs, but they usually make query answering intractable in data complexity, even in very restricted settings.