I wonder how often a (widely expected-to-be-true) mathematical hypothesis fails, versus how often it turns out to be actually true, versus how often it remained unknown so far.
If a hypothesis is widely expected-to-be-true, then there needs to be a good evidence towards it, which in turn means that it stood unresolved for a long time. That's why probably most of them are still unsolved.
Finding a counter example is often easier (in those cases that fail) instead of a proof.