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Silicone will do that.
Silicone.
I would use a hot glue gun.
Polyurethane construction adhesive like Sikabond Construction Adhesive will glue cork to marble. You could also use a hot glue gun.
Use contact cement. Home Depot sells Weldwood brand. I'd use the gel type, it's a little easier to handle with the cork. Get a disposable roller or brush for the application of the adhesive.
You should use a spray contact adhesive. 2-4 available in Home Depot.
there are sticky pads use those
TiteBond 2 will do the quite well, PL300 works also, but is thicker.
contact cement, hot glue, but I would use very small horseshoe nails, or you can make some out of a paper clip by cutting small pieces of it and making them into a U shape, then push them in over the ring and into the cork.
300lb watercolor paper, gessoed hardboard panels (preferred)
You can use any nail glue found in drugstores or you can use krazy glue.
Cork does not need to be glued. If it is compressed in the bottle neck, and then made wet, the water makes the cork expand. Have you tried super glue or hot glue gun? Not sure what you are trying to do, but maybe an "outside the box" soloution would work for you like straight pins. Whatever sticks out of the back side could be bent with pliers to lie flat against the back (like brads).