look at the initiative right-to-housing.eu they are collecting signatures to change that in the whole EU

A lot of investors like to keep their flats empty to reduce supply in lower income brackets. They use this to force people into flats they can't afford so the investors can get more money.

Uh-huh. So by foregoing rent they make more money?

Let's actually think this through: let's say that there are a bunch of landlords who each own multiple properties, and they all rent only half of them in order to, say, triple the rent on the rest (fake round numbers because it's a thought experiment). Now, what's the incentive for each individual landlord? If he's smart/greedy, he'll realize that he could rent out the rest of his units and get the 3X rent from them, too, because just his action doesn't move the market that much. So why the fuck wouldn't he do that?! And why wouldn't every landlord, acting insolation, come to the same conclusion?

The only reason that wouldn't happen would be if there were some sort of vast conspiracy between landlords to collude on the strategy, and if they were doing that they could just collude on the prices themselves and make even more profit!

The "they're intentionally leaving units empty to make more profit because reasons" argument just flat-out doesn't make sense. It's magical thinking to find a scapegoat.

And oh by the way, even if it were somehow true, that means your actual problem is a cartel and the solution is antitrust law. Even if you were right, it's still not a reason not to build more housing!!!