My shower is in the basement with a washer and dryer for my work clothes. I've always kept a spray bottle with 1/2 peroxide and 1/2 dawn for grease and oil on my clothes. Just early this week I used it on my shower, looked similar to what you described, and it worked better than anything else I've tried. I sprayed is on after my shower and the next night rubbed it down with a brush. If you try it be careful, the dawn makes the floor slick. I think it is the peroxide that does most of the work.