Time for nationwide rent control to keep prices from surging past UBI then. The argument against rent control has always been that investment will go where there isn't rent control since this maximizes profit for the investor. However if you give the investor no alternative then the law of TINA suggests that if they would like to have exposure to the U.S. housing market the only option is to invest in the U.S. housing market, and if the entire market is under rent control then there will be no winners and losers based on having local rent control ordinances in place or not.

Another solution is to invest more in our public housing stock. Private apartments could march up sure, but they'd be competing with the government building with subsidized rents designed to be affordable under the UBI plan, which would put downwards pressure on prices.

Are the immigrants receiving ubi?

If so how?

If not, how are they cheaper than somebody who only wants to work for fun?

Then the land value goes up to match rents and thus the ubi.

The ubi receiver still loses and needs other support

Land value taxes lower the value of land. The UBI recipient can conserve his share of land and rent it out to someone who needs it. That will result in an overall increase in economic efficiency and that surplus can in theory be used to buy products and services to meet the basic necessities of the recipient.

Alternatively, the recipient can run a homestead on his share of land.

You can increase the land value tax all the way to 100%. In the limit the UBI can be paid for entirely by land value taxes, which leave no surplus for land owners. It’s very easy for a UBI to end up benefiting only the non-working and land owning classes but it’s avoidable.