"""Removes the 'module.' prefix from keys in a state dictionary.

This function processes a state dictionary and removes the 'module.' prefix from any keys that start with it. The resulting dictionary retains the original order of keys.

Args:
    state_dict (dict or OrderedDict): The input state dictionary, which may contain keys prefixed with 'module.'.

Returns:
    OrderedDict: A new ordered dictionary with the 'module.' prefix removed from relevant keys, maintaining the original order.
"""