You have to wonder!
Dementia seems to be a situation in which the brain is not able to use energy the way it should. Brain-fog seems to be a lesser manifestation of the same nature.
One of the very first things I noticed when I first went on Atkins, over a decade ago, was that a few days into it, I woke in the morning with a clear mind.
Was it because I was no longer using wheat? It had been a long time since sugar was a significant part of my diet, but I certainly wasn't limiting starches before that. Up to that point, I had been a believer in the low-fat dogma that dominated the nutritional conversation since the '70s. The New Diet Revolution was astonishing to me from cover to cover -- trying it out made me change almost the entire contents of my pantry.
Was it some other food that I used to eat, which disagreed with me in ways I didn't realize? Maybe -- i'm more likely nowadays to wake up with brain-fog, when known allergens are rampant. Could have been dairy, could have been FODMAPs, could have been yeast-encouragers....
No matter what it was, it was blatant -- low-carb was as good for my brain as it was for all my other parts. Headaches became less common and thinking became sharper. Ketosis rocks.
If nobody has yet conducted a study looking at past "fog" problems and current Alzheimer's patients, I sure wish they would....