This assumes both A. everyone is suddenly just making more money and B. the supply of housing remains constant.
In terms of A, some people would drop out of their current employment. In all likelihood temporarily, but a lot of people are doing things they don't want to do because they don't want to be homeless. Likewise, it would almost certainly not be 97% of people making more money - that kind of progressive tax threshold would mean you don't even start taxing for UBI until 6 figure income.
In terms of B, a lot of areas of the country are currently unlivable for a huge percentage of people because there is no way to earn money to afford to live there. Living there in absolute monetary terms would be cheap, but absolute money is pointless if you have nothing coming in.
UBI reduces demand for housing in cities because anyone that doesn't want the added income from living there would now have the option to move away. City housing prices are extraordinarily beyond the flat costs of construction and maintenance - the scarcity of work means a scarcity of high demand housing in cities surrounded by deserts of unlivable land.
Finally, for all those dropping out of the workforce, the UBI is not an elastic income - it will be a simple binary can a UBI recipient afford to live in your apartments or not. For those who want an extremely reliable tenant who has a guaranteed ability to pay rent every month, meeting the UBI threshold becomes an extremely strong price ceiling on affordable housing.
Inflation isn't magic. Inflation happens insofar as someone is willing to pay the higher prices and nobody can produce supply to drive the price back down. Introducing UBI does neither to housing (especially when as a percentage of income UBI relative to the cost of home ownership would be a relatively small gain in income for most people, and almost all current homeowners / tenants).