Since you are not having any problem with starting the car when it is hot, then I suspect that your cold start injector may be intermittently working. There are methods to test it. This part is no longer available from the typical suppliers, but a few still have some in stock should you need to replace it. Try searching to forum for 'cold start injector/valve'.

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2006 BMW M5 "Heidi" @ 109,000 miles
2005 MBZ C55 AMG "Lorelai" @ 165,000 miles
1991 MBZ 300E "Benzachino II" @ 165,000 miles
1990 MBZ 500SL "Shoshanna" @ 118,000 miles

(On the hunt for a good used M103 engine as of 6/10/23, PM me if you have one to sell!)