It's pointless to try to prove anything to conspiracionists, they'll just tweak their stories and continue as if nothing happened. Anything that goes against their narrative is ignored and quickly forgotten, anything that appears to remotely vaguely confirm their beliefs is amplified and repeated ad nauseam.
Don't waste your time with them. You cannot reason people out of something they were not reasoned into.
Every conspiracionist beyond hope was once not quite beyond hope if you go back enough in time, and new ones are getting minted all the time and can still be saved. It's a worthy fight, even though it cannot be thoroughly won.
It has been argued that one saves people from going down the rabbit hole of conspiracy theories not by giving them the right facts, but by ensuring that they do not feel alone and isolated in society. Speaking with Flat Earthers can reveal that they accepted Flat Earthism simply because it gave them some feeling of community with other people on the internet, and the belief in a flat Earth itself isn’t actually the point.
Don't waste your time with them. You cannot reason people out of something they were not reasoned into.