[senseless]

No, it does not assume a fixed supply of housing. The law of rent revolves around the margin of productivity and already takes into account what you're trying to say.

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.