No it won’t, why would it?
If every single landlord knows with certainty (2) that every single tenant in their market has increased their ability to pay (1) by $1000/mo, then every landlord independently will come to the conclusion that they can raise rents by $1000/mo.
They know the tenants are able to sustain it, and they know other landlords know the same thing.
Most insidiously: the demand that will move most aggressively up the price curve is lower income people seeking better lower/middle income housing. So those prices will move the fastest.
Evictions went down for 3 months, then rose again back to their prior level after as rents increased.
This says public investment will just increase aggregate land rents to at least the cost of that investment. UBI is effectively a gigantic “public investment” that is everywhere. It’ll get baked into land rents, which will then propagate into actual rents.