Lock-free ConcurrentSkipListMap with complex object as value containing a list

Question

I am developing a program where I have multiple threads accessing a Map stored in the class Problem. This map, for the "value", contains an object (Slot) containing a name and a List of instances of Car.

So, my question is:

Since the first map, accessed by different threads, is thread-safe, will I need a thread safe list for the list inside Slot or am I safe with an ArrayList? Please note that Car is an AtomicReference and is the critical area where I use compareAndSet() on.

        public class Problem {
            ConcurrentSkipListMap<Integer, Slot> slots;
            //...
        }


        public class Slot {
            private String name;
            List<AtomicReference<Car>> cars; // does this need to be a thread safe list?
            //...
        }

        public class Car {
            //...
        }

A simple diagram that shows the problem:

Thank you.