FWIW, and adding to the nomenclature confusion, the US green card is often referred to as an LPR (Legal Permanent Resident) card.

As the holder of such a card, I have had no problems flying into Canada on AC. On the last occasion, no one even asked me about my status, and my traveling companion, who holds an ETA, was not asked either. We had not been asked for any details at OLCI and were issued BPs without issue. Is it possible that AC matches one's passport number against a database of some kind? If so, presumabvly UA could do the same.

Questioning on arrival by an immigration official was quite aggressive, though. He, too, did not ask to see any details of authorization to enter the country but was more interested in our reason for being there (a family wedding, as it happens, not that we could have proved that if asked to do so).