The concern is more that their children, and their children's children, are likely to be in the 1% as well, no matter what they do, or don't do.
http://time.com/money/3925308/rich-families-lose-wealth/
It is a very known fact, but I couldn't find better sources quickly, so this article has to do.
The concern is more that their children, and their children's children, are likely to be in the 1% as well, no matter what they do, or don't do.