The National Rifle Association (NRA) has John Alden's gun?

I saw a picture and caption on tumblr, the blog site, and had to check it out. Sure enough, have your browser find NRA Museums and click on "Guns" (top left) and pull down to "Galleries." Go right and select "Ancient Firearms." Right in the middle of the page, under "Significant Firearms in Gallery 2" is an entry marked "Mayflower Gun." There are 2 online videos about the gun. The museum curator does claim in the longer video that this Italian Wheelock Carbine was the personal gun of pilgrim John Alden and that he brought it with him in 1620. The basis for the claim appears to be that it was found in a "secret compartment" near the front door - still loaded - in the Alden house in Duxbury, MA in 1924. The Museum video says that the house was built in 1653, but some research pushes that date forward 30-40 years, with the Aldens simply reusing the timber from their previous home. The 17th century house was occupied by Alden descendants until it was bought in 1907 by Alden Kindred of America. Their web site has a history of the house, including a detailed PDF you can download, here.  Also, the NRA curator claims that John listed this gun specifically in his will. According to the GSMD's Alden silver book, part one, pp. 11-12, John Alden actually left no will. Citing a transcription of the probate documents, printed in the Mayflower Descendant (1910) 3: 10-11, which in turn cited Plymouth County Probate Records 1: 10, 16, there was an inventory taken of his possessions. It included "2 old guns."
This may well be pilgrim John's personal weapon but to prove it I would like to see some discussion of scientific testing done to the gun, rather than a discussion only of the gun's provenance. I am sure it was owned by an Alden, probably a John Alden since there were lots of them, and I am sure it was used a lot. Whether it came over on the Mayflower in 1620 with a young cooper depends on proof of its construction before that date.