Sharpen the kohl pencils with the nail clipper
A normal pencil sharpener is not suitable per se for sharpening cosmetic pencils with soft refill.
Unfortunately I did not have any optimal experiences with special sharpeners for kohl or eyeshadow pens - not even when storing the pens in the fridge. For a long time, therefore, I take a nail clippers and snap off parts of the wood around the pen tip with it. The pen can then be additionally made with the hand or toilet paper in a pointed shape. For me but unnecessary, because I also detect parts of the color substance with the clippers.
