I don't think they believe collusion isn't happening.

I think the argument above is that democrats are one of the drivers of building restrictions, leading to the ability to collude. If new entrants to the market were plentiful then the existing cartels would be undercut. Also, rent control puts a tight lock on the rental market by forcing landlords to keep their rents high lest they become locked into the low rents they may otherwise offer.

I'll leave it as an exercise for the reader whether to be on board with that assessment, but there is a reasonable argument to be made that a free-er market could actually benefit housing costs.

What this completely discounts is the existence of corporate landlords. Thousands of homes are being bought and kept empty to restrict that supply. It's ridiculous to destroy land used for other purposes just to build more empty houses and hope Blackstone et al. don't notice. It's also silly to hand those corporate landlords the right to jack up rents overnight as though they won't use it. The assumption that they'd rather have lower risk agreements over a longer period of time (e.g. lower rents now with slow increases year-to-year) is naively assuming that publicly traded companies will not attempt to maximize profits for the coming quarter.

> that are using Large Language models to keep prices high

Where do LLMs come into this?

what the h do LLMs have to do with this?