Have a marble staricase in a 1920's building that I'm trying to fix up.

Some of the steps are wobbling on the stringers, others just need some grout.

I have access to the underside and can see that some material that looks like grout was used to secure the marble to the wooden stringers. As it's mostly separated from the marble in many places, I plan on chiseling it off and replacing it. I don't know what material to use, and can't find instructions or information online.

Was thinking of using a plaster/grout mix with bonding agents added and just smearing it on from the underside to bond the marble to the wooden stringers and then, at a later time, grouting the side that's used.

Looking for suggestions and leads to information on how to rebuild/restore/clean and polish the marble steps.

TIA