Oh, yes, I also tried Zud which contains oxalic acid. In all cases I soaked the surface for 3-4 hours and for some cleaners I did overnight. So my feeling is that acids either do not work or very strong acids are needed where tile damage is likely. Given that this is a rental, I am looking for more gentle approaches.

When you place rubber-backed carpets on vinyl flooring, the vinyl will discolor. Permanmently. Nothing gets it out. White vinyl develops a yellow stain, haven't seen the effect on gray. Make sure the landlord knows that this predated your move-in.

My floor is definitely not vinyl though I am not sure what kind of tile it is. Would poultice work?

Try clear ammonia full strength. Wear rubber gloves and ventilate the room well. Apply a generous coat on the stain, leave it on for a few minutes, then wipe it off. If that doesn't work, nothing short of a stick of dynamite will remove that stain.