My 2001 E53 3.0i automatic X5 experiences a medium strength vibration when I have the transmission in drive, at idle and I have my foot on the brake. The vibration occurs whether the engine and/or ambient temperature is hot or cold. The check engine light never comes on. The engine is not throwing any trouble codes. I've had this problem at least six months.

The vibration stops as soon as I take my foot off the brake or I shift into park or neutral. The vibration does not happen at all while the car is moving. The engine is very smooth at all speeds. Handling and braking are also good. The is no smoke of any kind coming from the exhaust. There are no exhaust leaks (that I can find). The idle speed is stable.

This car has no mods of any kind.

The DISA/flapper valve is good (is working and does not leak), there does not appear to be any vacuum leaks, I've cleaned and checked the throttle body, I've cleaned and checked the idle control valve, I've replaced the air filter, I've replaced the fuel filter, I've replaced the ccv oil separator, I ran some redline fuel injector cleaner and I run high octane gas.

Does anyone have any other ideas for me to fix the issue?