That seems unlikely, because of competition. Not merely of identical units, but of all alternatives. We are not terribly supply-constrained on housing, nationally or globally (some markets certainly are, actually or artificially).
"Same way college costs rise to whatever loans are available."
Money that is available only to a single purpose is a substantially different dynamic than money that can be put to other uses.
It includes those cities which do not have housing restrictions. It includes any new units permitted inside existing housing restrictions. It includes increased sharing of housing (this is also impacted by other forms of "housing restrictions" but most people live in situations below maximum occupancy). It includes moving outside of cities entirely (made more attractive in the presence of a basic income).
To the extent that people choose these alternatives, rents will face downward pressure. They may still go up, but "the entirety of BI will simply become rent" seems wrong.
If I could earn a guaranteed $25k for breathing, I sure as hell wouldn't be where I am now, since I could go home and buy a good chunk of land, do some subsistence farming, and happily hack away and read at my leisure.
Secondly, if we go that far why not push for more public housing as well (can do it in a failing city like Michigan or build a new city) and let people create their own economies?
Or guaranteed minimum landlord profit.
Because rent will rise by exactly whatever guaranteed minimum income becomes.
Same way college costs rise to whatever loans are available.