> Yeah there’s a correlation but I doubt it’s causation.

There is. Density-misery death spiral.

You can usually check for correlation vs. causation by looking for counterexamples. Do you know a city that increased density and that resulted in cheaper housing?

I don't. And I checked the stats for all of the US, Western Europe, and Japan.

So far, it looks like the only way to decrease the cost is to reduce the city population.