Blow out the hole with dust, then use a gun two-part epoxy adhesive (as used for anchoring resin type wall ties) Fill each whole with resin (careful it goes off quick!) and push the studs in holding the shelf in place until the resin goes off. I think I would anchor several dowel screws similar towards the upper edge of the 7cm shelf, and then drill corresponding holes in to the stone, creating a slightly tapered hole wider at the base. You may need to create a hard edge along the wall where the bottom edge of the shelf will be trying to compress in to the wall. By using a fixing relying on tension you can discount stresses going downwards into the studs. What I would do is create a cantilever and get a good enough fixing that the TENSION on the fixing won't pull out. if you overload the shelf you are relying totally on the property of the fixing to resist bending. The fixings it came with are obviously designed to support the shelf with the fixing mainly in shear, but in my mind this is potentially weak if your wall is weak - the shear stress will be downwards through that tiny stud. This means if you put the fixing towards the upper edge you have a decent cantilever support (if you look at the fixings it came with you can see the smaller section isn't central on the larger - this creates a small cantilever and reduces bending stresses by increasing tension stresses - if you imagine putting a soft stud on the lower edge of the shelf you would expect it to bend rather than pull out). You have an advantage that the shelf is 7cm thick. The majority of the weight is being directed straight down. With the rads like the one pictured I would use feet like mentioned above if at all possible. I would discount the 2cm thick studs and fix in to the solid wall. Current task is a wall hung vertical weighing in a 68Kg dry to lathe and plaster.

Best fixings for lath and plaster walls