How would that work in a parking lot? There cannot be any blue tooth signals in the parking lot or my garage. Yet I have seen it do both. I never use the blue tooth proximity to unlock doors. I 100% use the unlock button on the key fob. I keep the app off on my phone too.
Bluetooth physical range is up to 330 feet so long as everything is to spec. If the truck is broadcasting bluetooth with more power than spec, or has a higher decibel antenna than spec, it could be even more sensitive than that. I do know that the truck has 4 Bluetooth sensors external to the vehicle for the purposes of proximity detection.
Sitting in my house I can see my car is connected to Phone Key, Sensor 1, Sensor 3, and Sensor 4. Not connected: Rivian Audio, Rivian Camp Speaker, and Rivian Wallbox (the charger).
Regardless of keeping the app connected or not, if your phone is bluetooth paired with the truck, the phone itself will be looking for Bluetooth beacons and so will your truck. Not to mention the Fob itself is also a bluetooth device.