You’re right; Gulf Arabs are more fanatical in oppressing women, both in terms of more extreme veiling like niqab, abaaya, and burqa, besides work and other social restrictions. However, Islamization, including hijabization, has advanced in Iraq, as well. This was utterly predictable. Remove a relatively secular tyranny in a devoutly Muslim country, and Islamaniacs are sure to fill the gap. Politics, like nature, abhors a vacuum.
I live in a country where men are discounted so many percentage points because of their appendage; who do you think they fight for? I have sons.