How do you convince someone they believe a lie.

Just found out one of my siblings has believed a lie about me for the past 5 years. They believe I did something that I did not do even telling our parents. I’ve been so confused about why my parents have been calling me a “liar” and a “snake” jokingly. I have anxiety issues and low self esteem so I figured it was because I’m awkward and don’t know how to be a person right. It’s also why I thought this sibling didn’t talk to me. That or they had their own struggles. Turns out it’s just because they all believe something untrue about me without even talking to me about it. For almost half a decade! I feel so betrayed. Nothing in my character would suggest I’d ever do such a thing so I’m confused. I even feel terrible my sibling went through such a hard time believing this information because I love them. I genuinely think they were manipulated by someone else. But Why would they believe this? How do I state my truth? At this point, does it even matter? Do I want to be loved by people who will believe whatever they hear about me with no proof? Where do I go from here? I love my family and thought they loved me… but this just makes me question everything.