I'm inclined to favor the air pump bearings beginning their "swan song" so to speak.

Since the intake mixture is fuel rich when the engine is cold, the air pump is designed to enrich the exhaust with air prior to going to the cat upon startup. An unnecessary piece of equipment on US-spec vehicles, european smog inspections require tailpipe sniffing within 60 seconds of vehicle startup.

The pump does its job in the fist few minutes and then shuts off...which may be why you don't hear the "whine" after that.

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2009 ML350 (106K) - Family vehicle
2001 CLK430 Cabriolet (80K) - Wife's car
2005 BMW 645CI (138K) - My daily driver
2016 Mustang (32K) - Daughter's car