No. Communism requires abolition of private ownership. However, the hyperinflationary spiral resulting from BI could assist a transition to communism.
Looks like you've identified the end game.
If it happens, I hope they do both, since the location adjustments will be even more rapid than the generic inflation adjustments, and it is possible that hyperinflation would just destroy the highest cost cities, rather than the entire US, before the program is scrapped.