Years ago in another life, I installed robotic storage/retrieval systems in Ford, GM factories. In some cases, we put them in offices and had to be careful when assembling with the grease, oil on the chains, etc.
When we did have a mishap, we used a bottle of Fantastik to remove the spots. We would soak the area and then use a paper towel to blot it and finally work the grease out of the carpet.
It always amazed me how well it worked.