I suppose Gödel's incompleteness theorems and Hilbert's second problem is along those lines, but I'm not sure Hilbert's problem was regarded as a conjecture.
http://www.openculture.com/2015/04/shortest-known-paper-in-a...
For 2000 years, it was conjectured that it followed from Euclid's other postulate. Lobachevsky, however, disproved it. (As with many 19th Century results, there are questions of precedence -- in part because Gauss claimed to have thought of everything first himself -- but his name is most commonly attached to the result.) Riemannian Geometry depends on the Parallel Postulate being false. General Relavity depends upon Riemannian Geometry.
I think there's some classic examples people pull out, but whether the relevant community actually thought them to be correct but unproven, I'd like to know.
I find this an interesting topic, the concept of we 'know' this is true but can't prove it yet.
I wonder if the idea of 'absolutes' sets back Mathematics sometimes. But maybe there are whole fields that rely on yet unproven but assumed true ideas?
Complex engineering for instance relies on 'this is probably true'. They make catastrophic mistakes, but they just pick themselves up and move on. Unless they are NASA I guess.
However, I'm not aware that he ever claimed to have proved the result, "wonderfully" or otherwise. :)
I recall being taught by my physics professor that Galileo never actually dropped two lead balls to test his theory - he reasoned that "heavier objects fall faster" made no sense because it implies that connecting two objects would make both fall faster.