Unfortunately, yes, scary reminder of the staingate period.
I guess it is in part some fatty substance that is left on the keys that make contact with the screen. Regular cleaning (for me, about once a month) with a very soft cloth, the ones used for glasses and camera lenses, to prevent built-up of dust and other particles that may become abrasive when cleaning is recommended. Careful circular motion will get rid of most marks; no fast wiping or pressure or you might damage the anti-reflective layer of the screen. I use a cleaning agent specifically made for glasses that helps, applied to the cloth before cleaning, not directly to the screen. But don't just use any cleaning agent, only ones that are designed for cleaning glass with protective/anti-reflective layers such as the ones for glasses or camera lenses.