I don't know about EK exactly, but seems like that most air carriers don't fly in Iraq and Syria sky if their destination or take off is not from Iraq. I think in Syria airspace only commercial carriers now flying are Syrian Arab Airlines and MEA (from destinations from Middle East to Beirut).
One day flightradar24 showed that Iraqi Airways passenger flight from Baghdad to Istanbul flew directly over Mosul, what is IS area now. Altitude was about 10 000 meters, but still..
Turkis Airlines, Austria airlines and many other airlines flies to and from Erbil daily. Erbil is about 90 km from Mosul and about 60 kilometer from IS area. I posted this post to the "Spectators Balcony" and it would be very intresting if some real pilot would comment it.