The lamp out indicator is only able to detect a burnt out lamp if the lamp is powered. If it is coming on as soon as the key is switched on it sounds like it could be a problem wth the lamp monitor itself.

-J

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1991 350SDL. 230,000 miles (new motor @ 150,000). Blown head gasket

Tesla Model 3. 205,000 miles. Been to 48 states!
Past: A fleet of VW TDIs.... including a V10,a Dieselgate Passat, and 2 ECOdiesels.
2014 Cadillac ELR
2013 Fiat 500E.