The screen had seen a lot of neglect but fortunately no severe damage, like cracks or gouges (right). It would need more than polishing, however, so I began with wet-sanding with a 5000-grit sponge, which proved to be not aggressive enough in this case. I switched to 1500-grit paper which is about as aggressive as I care to try on these plastic items. That seemed to do the trick, so I then worked my way through 2000, 3000 and finally to 5000-grit, on both sides of the screen. That got me to the cloudy finish shown below.
Again, beginning with the least aggressive approach, I clean/degrease the part, then lightly sand with 5000-grit to even the surface, remove imperfections, and help the Mother's Back-to-Black Trim & Plastic Restorer get a grip. Here, I've treated the upper portion of the part — the difference is dramatic.
The quality and look of your finished project is the sum of many large and small projects. These are two perfect examples of how all the little things come together to make the whole.