How to clean a white plastic iPhone case?

I recently bought a white hard plastic iPhone case. The white edges have been covered in some kind of laquer to give a matte appearance (which I really love) but being a white case, it gets dirty easily and has a grey tinge to it. I tried to clean it with acetone polish remover but it started to wear away at the matte laquer so I stopped.

The back of the case is a light blue mandala design which has also been covered in a laquer, except this one is super shiny. I'm afraid that if I wash the phone case in warm soap and water, it will ruin the shiny laquer on the back. How can I clean my iPhone case?

I clean mine with anti-bacterial wipes.

A Mr. Clean Magic Eraser. Wet the eraser and rub it gently over phone case.