Rent is based on demand and competition in the market, which is why it reflects consumers' incomes. That's the entire problem with UBI: it will just result in widespread inflation, in housing and every other market. The only way to keep rents from rising so rapidly is for people to be able to own the properties where they live.

(This does, indeed, fall under "some form of socialism" that the author deems magically "unworkable." Simply a tax code of affordable taxes for personal property - i.e. a home you own and live in - and high taxes for private rental property used to make a profit, with the intent that it should make financial sense to sell to an owner who will live in it.)