Rents in many places have gone up massive amounts in the last few years anyway. Sometimes doubling in a single increase. All without a UBI. Rents increases aren't tied to an tenants income like you sugest.
Also, your inflation would only be short term. Long term, after prices spike with sudden increased demand, the high prices will incentivize more supply, bringing prices back down into balance.
And there's nothing to say the UBI needs to be implemented all at once any way. It could start as small as $50/month, then ramp up over the course of a decade or two. That smother transition would allow the markets to adapt without major waves.
There have been wider, longer studies, and what your describing doesn't actually happen.
Gentrification is extremely local, tied to a specific area. UBI is tied to people. All people. In all locations. No mater where they go. The dynamics are couldn't be more different.