Removing Tiny Hard Black Specks in a Fiberglass Tub?

I have a friend that has hard black specks in her tub, almost like ink spots. At first I thought it was mold that was building up inside the jets of her jacuzzi tub, so I used a cup of bleach and a cup of dishwasher detergent powder, filled up the tub and ran it through the jets for about 20 minutes. That did nothing to the spots. We cannot seem to get them out unless we scrub and scrub, but we also don't want to ruin the surface because it is fiberglass.

Every time she bathes there are more, now they just keep building up. She said she can get them out if she uses her nail then her finger to scrub each individual speck, but then there is a smudge left and we need to scrub and scrub to get that out. We cannot figure out what it is or how to stop it. They aren't grease or mildew spots because they are very hard. We are getting exhausted with this problem and now there must be a couple hundred little spots. I have noticed there is also some in the toilet in the same bathroom that are very difficult to remove too. There are 3 other bathrooms in the house, but they don't have that problem. (The specks are in the master bathroom.)

By Carol N.