True, instead of just the low-income earners dropping out now that they have an unconditional deposit in their bank every week, now you also have a significant number of middle and high income earners consciously choosing to earn less because of the crushingly high taxes.
Well, you have the UBI increasing incomes at the lower end of the economy, which does apply upward pressure on lower-cost rents. But on the other side, you'd be decreasing the income/wealth at the top, which applies downward pressure at the high end of the market. The ultimate effect is not a straightforward calculation. I think the total size of the money supply is probably the most significant factor, and that wouldn't change if the UBI is funded by taxes.
I'm talking about the effects of wealth redistribution vs. money supply expansion. Redistribution doesn't necessarily increase prices since the total amount of purchasing power in the market doesn't change. It's certainly not as simplistic as "rent gets increased in proportion to the amount of the UBI" as you're implying.
Taking into account redistribution, the story is even more bleak, not less. Lower income housing prices will rise faster than higher income housing prices.