Liver flushing and high-fat diet?

Is Liver Flushing still considered productive if you're eating a high fat diet or a diet where the primary nutrient is fat (avocado, coconut, animal fats like butter, milk, suet, tallow, eggs, etc.)?

A diet like this keeps the gallbladder/liver working, especially if the diet has a lot of protein in it, which makes the Liver Flush seem unnecessary. Perhaps occasional supplementing with ES is good enough?