I was fiddling with it for hours last night and when I could get the smearing off, there were still some random specs that wouldn't move. When I could get those to move, it would smear again. I could probably keep trying and get it eventually, but at the moment it's more frustrating than an interesting puzzle to solve.
I ordered a new one from a different "brand" that looks suspiciously similar, probably both just rebadged. I will keep this one and try cleaning it again eventually.