I'm not aware of any studies done on the issue, but in my own personal experience brakeless fixie riders absolutely are being injured at a higher rate than other riders.

They run into other riders (who had to slow or stop for some reason), they run into cars when they would have avoided the collision if they had brakes, they go down steep hills thinking they can stop and then find out they can't (which often leads to really bad injuries if there's cross traffic or a barrier down below), etc.

I've been on a lot of large group rides with significant numbers of brakeless fixie riders, and I've seen all these things happen multiple times, when I haven't seen them happen to riders with standard brakes. (I've seen them run into cars, and run into riders in front of them, but in smaller numbers, and I've never seen one lose it down a hill like I've seen brakeless fixie riders do.)

Now, this is just my own experience, an anecdote, not a scientific study, but the difference seems significant.