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No, the 3-month rule only applies to visitors to the Schengen area. Since you have a residence permit, your passport and residence permit only need to be valid on the day you re-enter the EU.
This is the information all check-in agents will see as well. Specifically.
Passports and other documents accepted for entry must be valid for a minimum of 3 months beyond the period of intended stay.
- Passports and other documents accepted for entry issued to residents of Andorra, Monaco, San Marino, Vatican City (Holy See) and Schengen Member States must be valid on arrival.