Does a counterexample always disprove a conjecture? I'm sincerely confused on this topic (the title). I was wondering if a counterexample always disproved a conjecture. I'm leaning towards true because it's not really a counterexample if it evaluates to false... right? Any insight or help would be dearly appreciated.

A counterexample always disproves conjectures. A conjecture will suppose that something is true for different cases, but if you find an example where it is not, the conjecture must be modified to not include a particular case or rejected.