If they were truly good, then all they needed to do was believe that everyone would realise the truth sooner or later.
If they held onto such a hope, then it would become possible for them to prove themselves worthy enough to have the perception surrounding them change from bad to good so long as they continued to work hard and do a lot of good.
Even if some did gain a poor reputation, they would just have to hope that such matters would be forgotten eventually.
People would often be swayed by public perception in ways that could continue with such bad reputations following certain individuals.
Some of those things tended to stick in their memories, especially if they were big and memorable things.
Others could be quite impactful in the moment and a little while later, but things like that usually died down after a while.
People tended to forget about such things relatively quickly, especially if enough time passed without anything to remind them about it.
