The screen looked pretty bad at first blush but no severe damage, like gouges or cracks. I normally would begin the process by a light wet sanding with 3000 or 5000-grit, followed by buffing with two grades of compound. The sandpaper route can be a risky go, however, as just a bit too much sanding can often remove an invisible coating, just like clearcoat on paint — in this case likely an anti-scratch coating.
This proved to be a very effective alternative to sanding. I taped off the left half of the screen to show the change. The camera doesn't really pick up the full effect, but suffice to say it worked.