I wonder if the lightly acidic cleaning properties of Extra Virgin Olive Oil might loosen and help remove some of the heavier grey sediment??!! I have used it on Copper coins for many years to remove dirt, grime and encrustations and believe it would work on rocks as well. Even if it does not loosen and remove the sediment, it will probably stabilize the rock from decay and give it a nice appearance. If you decide to try Extra Virgin Olive Oil, afterward it's use, just rinse the rock with water then dry with a towel.
Frank