I have a couple of lawns that the wells used for irrigation have iron in the water. Sounds strange for a coastal community. At times the sprinkler heads will get hit by mowing guy, winds or for unknown reasons the water will hit the vinyl siding. After a while the rust stains start. I've used a product called "Rust Away" sold by gallons at Home De[ot. When spraying on the vinyl from a smaller spray bottle, the rust just goes away. The AI may be oxalic acid as Greendoctor said. It may work on the limestone as well. Worth a try as it's very inexpensive. Remember that limestone is calcium and is very soft which causes the stain to penetrate deeply and may take a few treatments rather than a prolonged abuse sorta like tattoo removal.
We all are merely gleaners from the masters who were here before us.