Landlords have pricing power because people have to live close to their jobs. If UBI lets people move freely, landlords lose their pricing power.
But there's a big gulf between "sticky" and "laws of supply and demand don't apply".
People on UBI will be much more price sensitive than those supplementing with wage income since the UBI dependent will be time rich and money poor. Landlords will have less power over them. They'll still have some power because of the stickiness.
You have no reason to believe people with new marginal income from UBI would move to lower COL areas. We have strong reason to believe they'd spend that money to move toward high COL areas. Evidence for this is the fact that people, when they have money, choose to live in high COL areas.
Your claim is, I guess, that people supplementing wages with UBI will move to poorer locations, because they are less tied to their jobs, and then the poorest of the poor (those who already live in bottom-percentile COL locations) will... move to even poorer locations?
Seems like a wacky argument relative to: "people will do what they literally always do with new marginal income, which is move to nicer areas, and since everyone is doing it simultaneously without new supply, prices will go up and nobody will improve their lifestyle."