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Added an Idea-section to this (previously stubby) entry — not meant to be definite, just what came to mind when finding that such a section was still lacking here:
In mathematics, a conjecture is a proposition which is expected to be true, hence expected to have a proof, but for which no proof is (currently) known.
Hence being a conjecture is a sociological aspect of a proposition, not a mathematical aspect: Once a proof (or else a counterexample) is found, the conjecture ceases to be a conjecture and instead becomes a theorem.
It happens that conjectures remain unproven while being perceived as trustworthy enough that further theorems are proven assuming the conjectures – in this case the conjecture plays the role of a hypothesis in the sense of formal logic.
For example, the “standard conjectures” in algebraic geometry serve as hypotheses in a wealth of theorems which are all proven (only) “assuming the standard conjectures” (cf. e.g. arXiv:9804123).
In other cases the term “hypothesis” is used synonymously with “conjecture” – e.g. for the homotopy hypothesis (key cases of which have long become theorems) or the cobordism hypothesis (on which a proof has famously been claimed but not universally accepted).
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