That SHOULD cover you for ILL... LEOs on that side write their own laws. FWIW handguns are illegal in chicago so broken down or not, you're going to be in a world of hurt.
Federal peaceable journey laws will protect you in the long run if you are passing thru chicago and it isn't your destination, but again, ILl LEOs (especially in sickago) do what they like and while you might beat the rap you won't beat the ride.
Just make sure you never consent to a search either. Locking it (unloaded) in a glove box or center consol should cover you. Also with it locked they can't do a "safety check" on it without either your consent or a warrant.