Not exactly sure what solution to use for it, it looks almost like a silt that's on it. I usually start with a soapy warm water soak, them move on from there. I heard windex does well with clay's and such, maybe try a little on a small section and see.
You can try a bit of vinegar to see if the white crust is calcium based first, hopefully it is, before moving on to stronger solutions. Chalcedony is pretty resistant to most acids. Other than that you can try buffing it with a dremel.