Upvote:2
The inflatable part, even if it works above 8,000 feet, may not be designed for the rapid drop in surrounding pressure in the cargo hold of an aircraft. It could be damaged by over-inflation.
To be on the safe side, I suggest modifying your puzzle pack. Get a cardboard tube at least as wide as the felt. Roll it up around the tube from the free end of the felt, and deflate the inflatable tube. Secure with bungee cords or in any other convenient way.