Perhaps try an acidic thick toilet cleaner with a soap in it or lemon juice with a squeeze of neat fairy liquid ( Fairy is the best as it does not have palm oil in it to any great degree unlike some lesser wash up liquids ) give it a gentle scrub , leave on a while give it another scrub and then with a warm water spray and an aqua type vac dilute the acid and soap as you gently scrub the stone again & suck the solution of muck and stuff off the slab .
Whilst having the central heating flood repairs done I had to do the whole kitchen floor & hall topping whilst sat in my wheel chair by the above method . For when the so called " builders " took the old oil central heating boiler out they dropped and left years of thick black greasy carbon soot ( about half a standard bucket full ) across the kitchen floor that was to be covered in glued down carbonised bamboo flooring. To add insult to injury they had walked it all over the empty bungalows concrete floors for three days before I found out ........... BTDS.
Obviously use poly sheet and tape to protect the surrounding areas.
The acid and Fairy will break up any grease & tend to bleach the stone , the fairy liquid will make the water wetter so loosening deep grease & non grease based dirt .
Using the clean spray with warm water will allow you to suck the crap out the stone pores as well as and neutralize the acid and soap.
If it is a water based deep stain you'll have to just accept it as " character" of the of your unique property .