Export certificate to take dog home from Italy required?

I've gotten conflicting answers on whether an official Italian Dog Export Certificate is needed to take our dog back home to the USA (EWR airport) from Italy after 5 weeks here. (note that this question is not about an EU pet passport or the CDC Dog Import form required to take the dog back home, I'm just asking about an Italian Dog Export Certificate)

When we visited last summer, on the way home the airline didn't ask for anything but the same USDA certificate we used on the flight from the USA to Italy to verify the rabies vaccination, and they asked us if we had filed the CDC dog import form.

This time we registered him with the ASL because we plan to become long term residents, and when he learned we'd be returning to the USA, the vet at the ASL said that Italian Law requires an export certificate to leave the country and we absolutely have to have an export certificate or the airline will not allow the dog on the plane. He also said we'd need a new appointment to get the certificate since he can't issue it without a separate appointment.

The US Embassy in Italy also says the export certificate is required: https://it.usembassy.gov/embassy-consulates/rome/sections-offices/foreign-agricultural-service/pet-travel-faqs-italy-u-s/

Our vet said "Yeah, some airlines require export certificates so we should get one, sometimes it depends on who checks you in at the airport whether or not they require it".

Our dog walker (who has many expat clients that bring their dogs to Italy frequently) said "I've never heard of anyone needing the export certificate, some get it, some don't", but as far as she knows, no one has been asked for it, and in about a half dozen trips back to the USA, she's never gotten the cert to take her dog back.

I asked our airline about it and they said they only require the same paperwork we used to bring the dog to Italy (i.e. the USDA cert) plus a statement of good health from our vet within 10 days of travel.

I've searched posts here and on the internet and have seen the same conflicting information, some say you need it, some say you don't, some say "it depends".

We'd like to avoid an unnecessary appointment at the ASL to get an export certificate if we don't need it.

Does anyone have a definitive answer (ideally with links to an official Italian source)? We're inclined to trust our airline since we flew home without the export cert last time without a problem, but would like to hear if people have been denied boarding without one.