Yes the liver does have a clearly defined 24hr rhythm. It varies from "food expectation" to detoxification cycles.
Liver bile and gallbladder bile is very different. The liver produces bile at a steady rate 40 mL per hour. It is made in a group of cells in the liver that filter the portal vein that it rich in nutrients and toxins from the intestines. These cells produce bile together with toxic waste and these flows into the bile ducts. (Note that the ducts don't produce bile) The cells lining the liver bile duct modify the bile but adding water bicarbonate and removing some bile salt. Liver bile is 95% water.
When you consume the flush potion and it enters the duodenum it triggers a lot of activity, but let us concentrate on the acidity factor. Stomach chyme is extremely acidic and the duodenum is the place where it is neutralise. It does this with sodium bicarbonate which is obtained from three sources.
1. The pancreas
2. Glands in the duodenum
3. Increasing bicarbonate secreted into the liver bile ducts by the cells that line the ducts. This swells the flow of liver bile. It is the increase in flow that cases the flush in the liver. There is a subtle difference between pressure and flow. The pressure in the biliary system is regulated by a valve and it is constant unless the gallbladder contracts when there is a sudden increase in biliary pressure.
Now for a word of warning: the Liver Flush has been carefully worked out to minimise risk. When the body enters the maintenance part of its cycle it relaxes the valves and muscles that normally control the flow of fluid in the digestive system. The "migrating motor complex occurs 2 hours after the last intake of food and last for 2 hours it allows solid material, like gallstones, to pass smoothly through the digestive tract without fear of being snagged in the terminal ileum. The Epson salts paves the way by relaxing the muscles in the gut and providing an hyperosmotic saline laxative.