This can perhaps be considered more of a meta-conjecture than a conjecture: Hilbert's program, [https://en.wikipedia.org/wiki/Hilbert's_program](https://en.wikipedia.org/wiki/Hilbert%27s_program).  The conjecture would be: that set theory (or some set of axioms suitable for doing math) can be proven consistent.  Gödel's Incompleteness Theorem disproved this conjecture.

I don't have a reference, but I have the impression that, at the time, Hilbert's program seemed attainable, and Gödel's result came as a surprise.