Just an FYI. Nail polish remover is essentially acetone, which will remove most adhesive without too many issue, but be careful on how and where you apply it (And it might require more than one application), because it can leave a residue which could attract other dirt and particles that could sticky between the lugs and links of the band.
So use care when applying. I would soak some on a Q-Tip or a foam tip brush and apply only to the effected area. If it removes it, I would then follow with a bit of rubbing alcohol and water, mixed 50-50 to remove all of the trace elements. Many multiple passes if necessary, but don't let it dwell on the surface too long at any one give time to try and remove it all at once. You need to let it work by dissolving a layer or two at a time if needed, and not soak it totally for an extended time to ensure no damage to the metal or surrounding attachments.
Sometimes adhesive becomes brittle or crystallizes when it gets old, and nail polish/acetone might not be strong enough to remove it completely. There are other non-caustic/acidic options for adhesive removers if needed, (You can Google then), but stay away from OTC items like Goop Off, etc, as they contain a high amount of enzyme-based, acetone-related chemicals and sulfides that could etch the stainless and would certainly eat away at any AR coating on the crystal if it contacts the surface for too long.