"Are there any conflicts among Anabaptist groups over cap style head coverings vs. hanging veils?"
I don't like to speak for Anabaptists at large because I have been pretty removed from mainstream Anabaptists most of my life time, so I will just tell you my personal experience and leave it at that.
I grew up in a church that wore the cap style head covering. Back in the era of my childhood, cap style coverings (with strings) immediately identified you as "Beachy Amish Mennonite". Any other type of head covering instantly signaled some other Anabaptist affiliation.
I remember as a youth when this began to change. You could no longer visit Calvary Bible School and point out all the people from Lott, TX because they weren't the only ones wearing veils anymore.
As a young married, the question of allowing veils came up at our church. It was vetoed after careful consideration. There was a pretty strong push for uniformity and not everyone was ok with changing to wearing a veil. It was during this time that I had my first eye opener on the subject. There was a young couple coming to our church at that time and the wife came from a church where the women didn't all wear the same type of head covering.
One day, this lady spent a day sewing at my house and we got on the head covering subject. I, in all my young wisdom, trotted out the popular argument, "Isn't it confusing to the community if everyone wears something different?"
I don't remember exactly how she answered but something of the idea that, no, it wouldn't be confusing because it is the principle of covering that is important, not so much how each person chooses to apply it. Mostly I remember that it was a new idea to me that uniformity isn't necessarily the only way.
Years later our church changed to allowing the cap style covering and the hanging veil.
I would say there has been a huge shift in recent years across Anabaptist churches to embracing different types of head coverings. I'm guessing though, you would still find some conflicts about the issue depending where you would go.