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ANSWER: Based on the information you've provided me, there is no reason why he would not be allowed to board the flight to Mexico, as it pertains to immigration documents.
Permanent Residents of the United States can travel to Mexico by air with their valid passport and valid green card.
I do not know why the airline would tell you that, but in terms of documentation making him eligible to travel to Mexico as a US permanent resident, there does not appear to be an issue.
Sounds to me like they were being overly cautious.
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