Yes, there will likely be that phenomenon, but will it occur faster than the approx 2% level of optimum inflation?

>>The only real solution there is for governments to look at social housing, and also to try to produce A glut of house building.

Creating a universally-available baseline lodging situation for everyone is certainly a public good that would yield a LOT of benefits from eliminating homelessness (benefiting not only the homeless but also everyone who their problems affect) to promoting family stability.

Whether the best way is to incentivize a glut, subsidize social housing, or just provide a housing stipend for anyone in need, another system, or some combination of all-of-the-above should be subject to study and experimentation.