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The Grinch may not steal Christmas at American Airlines, after all.

American says the staffing snafu caused by a glitch in its pilot scheduling system is nearly resolved, though the pilots union disputes the company’s assessment. 

The latest back and forth comes just a day after the carrier’s pilots union took to the media to warn that the error had left nearly 15,000 December flights without full cockpit crews around the busy Christmas and New Year’s travel days.

American acknowledged the issue, which essentially let too many pilots request time off during the second half of December. But the airline added it expected to be able to resolve the issue without the major holiday chaos that the Allied Pilots Association (APA) warned of.

Now, the company says the number of flights with pilot staffing issues has been slashed to just a “few hundred” and continues to drop.

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“Out of the 200,000 flights American will operate in December, only a few hundred are currently unassigned to pilots,” American Airlines spokesman Matt Miller said to USA TODAY’s Today in the Sky blog. “That number of open flights continues to decrease thanks to our pilots who are stepping up to the plate and picking up trips to ensure customers are taken care of.”

In addition to regular pilots signing up for those unassigned flights, Miller said the airline has “more reserve pilots on hand in December than normal months and they provide us with the ability to fly many of the trips that are currently uncovered. We have not canceled any scheduled flights in December and will continue to work to ensure both our pilots and our customers are cared for.”

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Still, the APA is disputing American’s assessment, saying that it believes “thousands of flights are still listed as unassigned.” 

“We remain seriously concerned about the potential for significant schedule disruption for our passengers, pilots, and fellow employees during the critical holiday travel season,” the union said in a Thursday afternoon press release

APA’s Thursday statement comes after it warned on Wednesday that major chaos could result during the busy holiday period because of the glitch.

“The system went from responsibly scheduling everybody to becoming Santa Claus to everyone,” APA spokesman Dennis…