Mathematics several times has conjectures of form

$$mathsf{Statement A}impliesmathsf{Statement B}.$$

In such cases falsity of $mathsf{Statement B}$ implies falsity of $mathsf{Statement A}$. However since falsity of $mathsf{Statement A}$ does not imply falsity of $mathsf{Statement B}$ it might be that disproving $mathsf{Statement B}$ might be the easiest route to disproving $mathsf{Statement A}$. However a direct disproof of $mathsf{Statement A}$ might reveal something else not directly revealed by $mathsf{Statement B}$ without falsifying $mathsf{Statement B}$. Are there known good examples?

For example Merten’s conjecture implies Riemann Hypothesis. However I am not sure if falsifying Merten’s conjecture revealed something different from a disproof of Riemann Hypothesis since the latter’s status remains unsolved.