I've never seen any research, but the purpose is more to lock the heel and foot in the shoe. Look at wrestling shoes- they're actually designed to have tons of ankle flexibility but lock the sole to the foot for maximum grip. If it really prevented ankle roll then you would see high tops in all sports with lateral movement where ankle/knee injuries are common such as soccer, American Football, trail running, etc.
For hiking boots it is quite different as boots extend far up the ankle past a normal high top and are built to provide more ankle stability/shoe stability.
A good thing to look at are ankle braces- they all wrap underneath the foot and then the ankle lacing/straps cranks over this. The wrapping is what prevents rolling.
I have no ligaments in my right ankle and shit is f'd. I was really curious about this for skating when I came back, but rolled my ankle fairly bad with Blazer Hi's. For me stability comes down to sole design and arch support. Put on a shoe and try to roll your ankle- a good shoe will snap you back to center. A bad shoe will tip you right into roll.