Can the western honey bee remove Varroa mites from other bees in the hive?

The varroa mite originally came from East Asia where they were a parasite of the Asiatic honey bee (Apis cerana). Over the millenia in which these creatures have lived together the Asiatic honey bee had developed several behaviours to keep the varroa under check, one of which is to pick mites off each other.
The Western/European honey bee (Apis mellifera) has only had to deal with varroa for a very few decades -- a mere blink of the eye in developmental terms -- and has not developed these behaviours, so although physically capable of removing mites from each other, they don't.