There are plenty of campaigners on the left - the UK Green Party is an example - that openly campaign for a BI in addition to most of the state benefits that already exist. And when it comes to implementation details, it's quite hard to believe that many left wing proponents actually would take the "well if the BI isn't enough for you to afford your social housing once we sell it off, maybe you should just move" route.
There are plenty of campaigners on the left - the UK Green Party is an example - that openly campaign for a BI in addition to most of the state benefits that already exist. And when it comes to implementation details, it's quite hard to believe that many left wing proponents actually would take the "well if the BI isn't enough for you to afford your social housing once we sell it off, maybe you should just move" route.