Even if that hypothesis were true then there is a question what is causing the immune system to become overactive? Are humans built such that some increase in the immune activity caused by some dietary change (i.e. more animal fat) were supposed to kill us by heart diesease where as a decrease were supposed to kill us by infections? I doubt! If they want to prove that atherosclerosis is caused by the immune system gone amok then they must find out what exactly does cause the immune system and the ensuing endotheliarl repair to go amok! Immune system is homeostatic - it stabilizes itself.
So when you want to know how sugar provides energy to the body do you also want to know each and every chemical process in the body that allows the sugar to be utilized by the body starting with digestion? Or do you accept the fact that sugar provides energy to the cells? Or do you want to debate that even further that it is really not the sugar but rather the ATP generated that is providing the energy to the cells involving numerous other reactions that must be verified before you accept it?
On top of that where is the evidence that there is even an overreaction by the immune system to begin with? So I guess since you want to claim that the process involves an overactive immune system that is where your evidence needs to start.