I just noticed you are in waukegan: I drive past frequently. I have some high-end soldering gear including the proper machine to desolder PCB soldered in relays, i could work in a stop on my way to/from chicago to help replace that relay if you determine that's the problem.

a search on the exact part # showed them for sale on eBay, but can also order from digikey, probably about $5-10.

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