My intuition is that housing prices will go down assuming that population doesn't change too dramatically and assuming that while AI is slowly taking off there are also sizeable takeoffs in energy production, transportation and automation of physical labor. I assume the price of land will be mostly evenly distributed as logistical costs of building and labor drop. The exception might be areas in the world that are especially sought after for AI infrastructure.
This is motivated by How to make money off of AGI. One of the predictions made there (and elsewhere) is that interest rates are going to go up a lot as growth starts to accelerate, and thus it's a good move to take a loan at low fixed rates now. However, the only way mortals can take this kind of loan is through a mortgage, and this is in fact an investment in real estate - if the asset price drops it doesn't matter that rates rose, you'll still lose money.
There are many factors influencing the prices in opposite directions so it's unclear to me what the net effect will be.
Factors influencing prices down:
Factors influencing prices up:
The regulatory response also seems pretty unpredictable to me, and it could influence all of these.