Not exactly Mensa card holders.
Family made a serious mistake when it handed over the coins for authentication. No doubt the offer to authenticate was a ruse.
Possession is 9/10th of the law and the family is most likely SOL.
IIRC, the coins were sent to the Mint for verification of their authenticity prior to ownership transfer. The 1933 Double Eagle is quite a story among the Numismatic community, but it is by no means the only case of coins which were minted but never released ending up outside the walls of the mint.
For a basic background of the coin and its history, you can check the American Numismatic Association's web page here.