The way I understand the "trickle down" idea is that the global economy is not a zero-sum game and when rich people build something, the poor indirectly take advantage of it, even if the wealth repartition curve doesn't change. For example, a rich person may use his wealth to make cheap computers, which sell well, so he becomes even richer, the poor don't become richer, but now they have computers, so in the end, even if wealth inequality didn't change, or maybe even became worse, the poor are better off, thanks to computers being cheaper. If we didn't let the rich entrepreneur get rich, there would be no cheap computers and while the poor may be wealthier on paper, they still won't have computers.
But if we only care about money, then yes, of course inequality causes inequality, and if the "trickle down" theory says otherwise, then it is simply illogical, just as its debunking is circular reasoning.
The study is not worthless, it is an interesting observational study, but I don't see how it disproves the more interesting non zero-sum version of "trickle down". For that it should also use indicators that are not money, like health, education, etc...