"Every single adult member receives a weekly payment from the state, which is enough to live comfortably on"
would mean a different value in San Francisco than in Grover, NC.
If the government adjusted the basic income to account for regional cost-of-living, I imagine it would be popular to maintain a fake residency in an expensive area while actually living somewhere cheaper.
If the basic income wasn't adjusted for different areas, you would probably see some areas become rich elite enclaves while other areas become basic income ghettos.