I kept my hair while undergoing paclitaxel chemotherapy using the cold cap. It thinned, but not too badly. To my surprise, after the chemo finished (within a month) was when most of the shed happened, including all my eyebrows and eyelashes! The eyelashes grew back, then fell out again and eventually returned to normal months later.
I treated my hair really gently during and after chemo. Minimal washing, gentle combing supporting the hair so it didn't pull, no blow drying or colour for a few months after. I was able to look pretty much like my normal self so the extra tlc was worth it to me. I did look a bit less 'groomed' during that period tho.