public class MyClass{

    private final Map<(MyObj, Map<String, List<String>>> conMap = new ConcurrentHashMap<>();

    //...

    //adding new record into conMap:
    private void addToMap(MyObj id, String name, String value){
        conMap.putIfAbsent(id, new ConcurrentHashMap<>());
        Map<String, List<String>> subMap = conMap.get(id);
        synchronized(subMap){                            // <-- is it necessary?
            subMap.putIfAbsent(name, new ArrayList<>());
            subMap.get(name).add(value);
        }
    }

    //...

    public void doSomthing((MyObj id){
        List<Map<String, List<String>>> mapsList = new LinkedList<>();
        for(MyObj objId: conMap.keySet()){              // <-- seems unnecessary..
            if(objId.equals(id)){
                mapsList.add(conMap.get(objId));
            }
        }

        for(Map<String, List<String>> map: mapsList){
            synchronized(map){                       // <-- is it necessary?
                //do some stuff..
            }
    }

    //...

}

if(conMap.containsKey(id)){
    mapsLists.add(conMaps.get(id));
}