These assertions raise doubts in my mind regarding the study as a whole. Fat and sugar (a carbohydrate) account for two of the three most common macronutrients, the third being protein. Eating a diet that's low in both fat and carbs would imply a mostly-protein diet which can lead to illness[0].
The further implication that a particular macronutrient ratio could cause obesity is particularly concerning. Weight gain/loss is determined strictly by whether someone consumes more or fewer calories than their body burns in a day. The ratio of macronutrients may effect other health indicators like lean body mass, but a diet of 100% sugar and fat will still lead to weight loss if the total caloric intake is lower than what the body requires thereby requiring the body to burn muscle/fat for energy[1].
While the researchers have found a correlation between memory function and obesity, they appear to be making spurious claims that the foods the obese people eat are a contributing factor. There's no indication they made any attempt to control for the fact that healthy people also regularly enjoy ice cream and bacon, just at a lesser rate than those who are chronic over-eaters and therefore obese.
[0] https://en.wikipedia.org/wiki/Protein_poisoning [1] http://articles.latimes.com/2010/dec/06/health/la-he-fitness...