Cleaning a circular polarizer

[PopsPhotos]

In all honesty, I haven't found many things that a microfiber cloth and a little "laser breath" can't remove.

That is my experience, also. However, when that doesn't work, one must use other, more drastic means, which don't include more pressure on the surface. You don't want to "scrub" the surface, just wipe gently with the wetted microfiber cloth until the sturborn thing lifts off.

Pops

[Ken MT]

From Hoya filter website
"CLEANING YOUR FILTERS:
Due to their high precision, filters should always be handled with care and kept clean whenever possible. Filters should be cleaned gently with just a lens tissue or soft cotton cloth.

Never use any chemicals, such as lens cleaning fluid, on your filters, as these can damage the coatings. If any stubborn stains occur, these can usually be washed off with some clean water and a soft cotton cloth."

[Gingerbaker]

Maybe hot water from the tap to melt the grease away?