From what it looks like in that film, yes, it becomes their identity.
Gradually their friends and family no longer see John Smith, but John the flat earth guy. If they then reject their flat earth community, regular society is hesitant to accept them back.
Some of the people in that film, I was looking at them and thinking "if it wasn't for flat earth, you'd be alone and definitely not famous". And it is truly sad to see people building communities around and worshipping a poor understanding of science.
It reminds me of when butters from South Park sets up a church in his local DVLA office 😂
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