I haven't tried "this method to cleanse the liver" - but for 4 years now betroot (a small one or half a big one) is part of my vegetable juice, that I prepare nearly daily from fresh vegetables.
I haven't noticed anything like a special Liver Cleansing effect. Liver Flush is the best way to cleanse the liver.
Of course fresh juice is a source of miracles - that's the reason I'm having it daily - and their migth be an effect on the liver, who knows...!