Lifeproof left marks on iPhone glass

I was using the Lifeproof fre case on my iPhone 5S. It has a thinner piece of plastic over the home button to still allow for Touch ID input. I stopped using the case and noticed that the edges of this plastic left some marks on the glass.

What is strange is that I can clean the screen with a microfiber cloth and the marks appear to be gone. If I smudge up the glass a bit, the marks (in a ring formation) return. Any idea on what this is and, more importantly, how I get rid of it?