Just slap a massive fine (I mean tax of course but that is just semantics) on any profits from a second or higher house. And dont start with the "you can't change the rules of the game in play"; sure you can, you just dont like it.
I always wondered why law didn't rely more in mathematics to regulate such dynamics. It is trivially easy to describe a formula which gives you a tax that doesn't impact the many have-nots while preventing perverse accumulation and still leaving enough slack in the middle to incentivise people to take part in a market. And anyone who disagrees: you can tweak the formula to any cutoff point, you can make it grow linear, logarithmic or exponential etc.
That being said, things like your solutions should always come with thorough check by a red team. E.g. what about the rich guy just forming a new company per property to evade that tax? So maybe it is smarter to bind that to an actual owner person. How will the shape of houses change if their count gets important (that could be how we get these cyberpunk-megabuildings)? Maybe it should be about rental units instead (or some other clever definition)?
Sometimes I wonder if the best solution wouldn't be to prevent companies from owning housing alltogether..
The thing is that the interest rates need to stay higher for a long time (many years). Many home owners and speculators have fixed interests on their loans. It is not until they have to be renewed that they feel the effect of the higher interest rates. So unfortunately it will take time.
Yet prices keep going up and up and up and up.
Just slap a massive fine (I mean tax of course but that is just semantics) on any profits from a second or higher house. And dont start with the "you can't change the rules of the game in play"; sure you can, you just dont like it.