remember that the body can generate glucose out of even fats, if necessary, so if it wants to, it can scavenge stores for its glycogen from any food at all, no matter how low carb it is.
So I don't think it is a case of starving it of ALL glucose, but rather keeping blood glucose steady enough that it relaxes its panicked belief that it needs to hold huge reserves.
I feel a bit sorry for our livers. They took years to learn that we stuff carbs, and expect them to hold the bursting glycogen bag for an emergency - and then we have pseudo emergencies every d*mn morning, so it learns to dump bigtime. But then we suddenly change our minds and blame it for something it learned to do to help us - AND we want it to change a modus operandi that took years to establish... poor thing. It must be VERY confusing!