See if you can get into module like HVAC. If you can but not EGS, then get some deoxit pro gold and clean the contacts at the egs module. Your symptoms match bad wiring or bad connector. (or internal fault).

You can open up the EGS and look for corrosion inisde. You may have to drill out rivet I can't remember if it was DME or EGS that wasn't screws. (you can replace with screws).

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)