Can Eating a Low-Fat Diet Improve Breast Cancer Survival?

I read several article versions of this study, and I am no wiser than before. The study notes that the women in the low-fat diet group weighed less than the others, on average. Isn't it entirely possible, for example, that the health benefits in the low-fat group were because those women maintained lower weights?

I eat a relatively low-carb diet, meaning that I eat a lot of veggies, some whole grains, legumes and some protein, usually fish or poultry. If I don't consume quite a lot of fat, I simply can't keep my BMI in normal range. If I have to cut the fat, mostly olive oil or the fat in the fish I eat, I would have to replace those calories with carbs and proteins, and eat a higher volume of food.