GKI has very limited application, there, especially for people who aren’t eating a substantial amount of carbs. Protein is the most potent inhibitor of autophagy there is, and somebody on a keto diet may be running fairly high ketones and fairly low glucose, with no reason to think autophagy is increased at all.
Coming from a carb-heavy diet, fasting results in the stomach emptying, the small intestine gradually emptying, the ‘post-absorptive’ phase occurring where insulin and blood sugar begin declining, and eventually glycogen depletion occurs. It’s reasonable for most people to assume that autophagy is ramping up during the second day of fasting, here, or the third day at the latest.
If eating keto, then glycogen storage should be much less, i.e. ~100 grams or so, versus 400 or 500 if eating carbs. I think it’s a “maybe/maybe not” thing if autophagy could begin to be up-regulated toward the end of the first day of fasting, there. Things are more favorable for autophagy starting out, since insulin should be lower and glucagon higher, but the main thing is for mTOR to be suppressed and AMPK to be activated, both these things essentially happening in the absence of nutrients.
It’s not totally either/or for tearing down protein structures and rebuilding - even if certainly in autophagy and scavenging protein, some amount of necessary cellular processes are still going on where protein is used.