What we usually see in cities all over the world is that increasing population density drives the house prices up, not down.
For whatever reason, people like to live around other people. The more people already live in a city, the more other people want to move there, and this creates an upward pressure on housing prices.
I disagree. Cities are in a endless race to build more housing. They need to continue building housing to keep prices temporarily lower, a metaphorical treadmill as more people will always move into the city. If a city stop building housing, prices will go up for sure because demand continues to increase. Cities of course always have more demand, but that is their nature as economic and cultural hubs, we have no choice but to try and match this demand.
Housing prices will not be low without building more housing, except your city has a specific reason that lowers demand, like lawlessness, economic ruin or some other disaster.
Which is meant to be balanced out by increased housing density like we see in Asian mega cities like Tokyo and Bangkok which have very affordable housing despite many many more people. By making it impractical or outright banning high density housing this is the situation you arrive at.
For whatever reason, people like to live around other people. The more people already live in a city, the more other people want to move there, and this creates an upward pressure on housing prices.