On the Wikipedia page for 'Anchoring', there is an example about thinking of a number and subsequently bidding (experiment by Ariely). What happens when you tell people with a high number that they are likely to make a relatively high bid, so they should try to make a relatively low bid? Do they make 'normal' bids or extremely low bids?
This shows that saying people can't rid their decisions of that influence is an overstatement. When you are consciously aware of the heuristics and the decision is taken over a period of time, you can downplay them.