Black/grey staining in bottom of toilet bowl with water softener?

[JoeSD]

We have a catalytic carbon + water softener system, running for 4 years now. Over the last year or two the master bathroom toilet has slowly developed this blackish/gray staining in just the bottom of the bowl, and now the other two toilets in the house are slowly developing the stain. Scrubbing with toilet bowl cleaner and a pumice stick does nothing to remove it. I thought water softeners were supposed to keep toilets nice and white? I have a Hach 5B test kit and test every month, the result is always 0-1 drops of solution to turn the test water from pink to blue.

Any ideas? I've enclosed three pictures one of each toilet with the varying degree of black/gray staining.
Guest bathroom toilet upstairs:

Master bathroom upstairs:

Downstairs toilet:

 

[Jimster]

I wish I could help. I am having the same issue. Often happens after a rain, but always on the hot side only. If I took pictures they would look the same as yours do.