I'm starting to wonder if there's any relation between derived private keys and their parent that can be exploited to eg. avoid searching for nearby keys and speed up the search. Something like trying to find random private keys by deriving all the paths but slower because of all the extra ECC math.
Since the keys are derived using a hash function and there is no relationship between the input of the hash function and the output of it, there is also no relationship between the master key and each child key that is derived from it that could be "exploited".