Landlords aren't in proper competition in supply constrained cities (most of them), so they aren't actually competing that much with each other. In reality, landlords are charging as much as the tenant can afford to pay (in aggregate). This means that the tenant will not be able to pay a higher rent, even if every other landlord also raises rent, so instead they become homeless (or move back with their parents). This happens across the tenancy market, so landlords are already charging as much as they can.

Lars Doucet has a good essay on this with empirical sources on AstralCodexTen: https://astralcodexten.substack.com/p/does-georgism-work-par... I believe he frequents HN as well.