I had a similar issue that also impacted the control of the passenger side mirror and door lock. The issue in my case turned out to be that a pin connector in the passenger door harness cooroded until it was gone. It appears signaling for all functions on the door came via that pin so after configuring a soldered in lead to bypass the failed pin All is well.

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2013 X5 xDrive3.5i Premium
2002 530I Sport Package
2001 X5 4.4i
2001 M3