What you're describing is the Climate Control's internal fan,it's a tiny one behind the left tiny grill and it stays on until X falls to sleep,exactly 16 min.
Keep in mind that you don't need to sit in the car while waiting for it to sleep,even if the door is open X is gonna sleep if there's no motion present in the cabin,if so it will take longer,but once it's asleep even if you wake it up with some motion it's gonna go to deep sleep mode within 30 seconds.