According to an Apple support document, the Pro Display XDR with nano-texture glass must be cleaned only with the dry polishing cloth that Apple provides. No water or liquids should be used to clean the glass.
Apple warns that Pro Display XDR owners should never use any other cloths to clean the glass, and if the included dry polishing cloth is lost, Apple Support should be contacted so another cloth can be ordered. There’s no word yet on what Apple is charging for replacement cleaning cloths.
My guess is that most cloths would get torn up and maybe leave bits of lint embedded in the nano-texture glass.