Laptop is depolymerizing -- how can I remedy this?

I acquired a ~16 year old laptop. The mat black plastic top (back of the LCD) is sticky. At first I thought the previous owner had stickers on the back that were removed. But that seems like a bad theory now. I rubbed it with a cloth and denatured alcohol and it only got slightly less sticky, but black residue came off on my hands and the cloth. This is apparently not adhesive.. it’s the plastic itself.

What’s my best move? I don’t suppose I can do anything to re-polymerize it. I don’t care about cosmetics.. I just don’t want it to be sticky and marking anything that touches it. One temptation is to put plastic film on it, like cling wrap. But that could just make a bigger mess.