Best advice is contact the authority on this in Chicago via email. Lay out what u just described. My experience is they will link you to the laws on this topic and explain then out for you a bit. Then save the email on your phone or print it.
I traveled to Iowa this year and was contacted by a sheriff. I have a Colorado CCW, we do share reciprocity with Iowa. The deputy wasn't convinced. So I showed him a email from the office of the state manager of safety. That prompted him to contact his superiors and I was then sent on my way.