My daughter in law managed a perfume shop and the optician often asked if he could have any drops of perfume left in bottles that they had left over. He told her that perfume can polish out very fine scratches with a little effort.
Also that perfume can get felt pen off sufaces etc., including leather . Tried this out when my grandson Ben wrote on my beige leather chair - it worked a treat!!
Worth a try!
Good luck
Chrisx
Might try that myself, my son drew us some lovely pictures on our double glazed french doors with a stone. I think that they are probably too deep to polish out.