My understanding is that all owners of would increase the rents/prices of their unimproved land, not just housing landlords. Everywhere you'd move is now more expensive by the same amount, because unimproved land is (mostly) fixed supply.
I do like the idea of a land tax though, and believe that it's a good way to fund BI (along with carbon tax).
The wages extractable from the least valuable land would rise by the same amount as anywhere else since basic income would affect everyone's incomes equally. Thus, there would be an excess over the basic level of sustenance and according to this law that excess MUST be swallowed up by rent. The status quo would resume for normal wage-earners, while land-owners would benefit even more than they already do, raising rates of inequality further.
Housing supply isn't fixed if you think people are willing to live in Wyoming - which they pretty much aren't.