Financial scarcity and cognitive fatigue

One statement that I find puzzling and a bit unconvincing is this: the poor make suboptimal decisions because they are being constantly cognitively fatigued and more importantly, their mental resources are being depleted to a greater extent than the rich.

Well, first of all, I'm not sure that the poor are being depleted more than the rich. The rich may be depleted by having to think about their business/investment problems which are computationally equivalent to (if not more than) the daily expense problems of the poor. Of course, if you ask the rich to think about their personal finances they will be depleted because it's simply not something that they are used to thinking about.

Secondly, maybe this is a better proof, suppose we strip away the rich's money and bring them to the same level as the poor (much like now) we'd expect the (formerly) rich to display similar behavioral patterns of the poor and be equally cognitively fatigued. If the economy is reviving which group, the formerly rich or the originally poor, will have a better chance to become rich? Certainly the formerly rich. Now, during such process of becoming rich again the formerly rich should be more fatigued than the poor because they not only have to think about their personal finances but strive and fight for any opportunity that will lead to wealth.