Is the majority of conjectures confirmed or rejected?
As a layperson I can think of several cases when conjecture was confirmed but none when it was rejected. For instance
http://www.scientificamerican.com/article/math-whiz-solves-a-master-s-riddle/
and
https://en.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem
Both cases are newsworthy regardless if the conjecture was confirmed or rejected.
How do I improve my question to be more specific?