If helpful for others, this is how I resolved the issue of lost AirPods case in a QTR aircraft.

1. Email to [email protected]
2. I received a response within 24 hours asking me for more details on the item (like serial number). Tha email also had the phone number of the Qatar baggage service office located in DOH, arrival hall, next to carousel 1: +974 40101812
3. At the same time, I could monitor location on Find My, I could see that my item was still in an aircraft as I was receiving location updates matching the flights of the aircraft I took
4. I called the baggage service number on one day, just after the aircraft landed in Doha. They sent someone checking the seat thoroughly and advised me to call back 30 minutes later.
5. Once retrieved, they told me how to get it back (transfer desk at my next connection in Doha or showing up in person at the desk in arrivals).