Final answer:
Franz Mesmer's claim about his mesmerism abilities that led to his downfall was healing powers.
Step-by-step explanation:
Franz Mesmer's claim about his mesmerism abilities that eventually led to his downfall was healing powers. Mesmer claimed that he could heal people through his mesmerism techniques, which involved passing magnetic fluid through their bodies to balance their energies. However, his claim was later found to be unsupported by scientific evidence and was seen as a form of pseudoscience.