ESPHome + Mitsubishi HVAC

If you're referring to the products from Mitsubishi Electric (which are the targets of the links I posted previously), they don't make systems like the ones you are referring to, where the air leaves the building and returns to it.

What they do offer, in addition to the mini-split style systems, are indoor air handlers which are connected to outdoor units which are similar to the ones used for mini-splits. Our home has two of these systems: an air handler in the attic and one in the basement, and two outdoor units handling the energy transfer with the outdoor air.

These systems do indeed use proprietary wireless thermostats, but the receiver unit for those thermostats plugs into the same CN105 connector in the air handler as it would plug into a mini-split indoor unit. The bulk of the protocol spoken over the CN105 connector has been reverse-engineered, and these ESPHome (and other) implementations can be used in place of a Mitsubishi thermostat with no loss of functionality.

As it turns out, the manufacturer's thermostats are not that intelligent (at least in the case of the MHK2), they defer all of the decision-making about fan speeds and inverter speeds to the air handler/outdoor unit combination. Many people, including me, have used ESP32s running ESPHome instead. I'm going to be experimenting with connecting both ESPHome and the MHK2 simultaneously, with the MHK2 having control but with the ability to monitor with Home Assistant (and obtain power delivery information which is not made available otherwise). Even the Mitsubishi 'cloud connection' unit also plugs into CN105.