In the end I tried to iron on the floor as you suggest Spicebar and it seems that it makes wonders. Here is a picture of the result:
I cannot say if it is really 1/5 of the original size but it is definitely at least 1/3 or 1/4. It made it quick to see if it doesn't mark the floor and apparently it doesn't. The only problem is that I didn't clean the floor properly hence there were some little impurities which got in the way of the ironing process. The landing pad has some little bumps at different places. Next time I'll try to move the force applied on the iron so it doesn't reproduce.
EDIT: I put it at max temperature for 40s. It was stuck a bit on the folded paper sheet but it was easy to pick it up so I guess it is around the right temperature for it. I'm looking forward to do it properly this time. I feel that it is going to be just wonderful after that.