I agree with all the answers/info thus far.
What you likely have is corrosion from improper storage, i.e. exposure to air, excessive humidity, and temperature swings. It won't hurt to try a short soak in acetone just in case there's an off-change the spots are organic. However, corrosion fits the bill better based on the pictures. Chances are the spots will be irremovable without damaging the surfaces. If the coin is important enough to you, an acid-thiourea dip is your only chance IMO. Sorry to say... I think the corrosion has already progressed too far to save the coin.
Also, like Jim said, the corroded staples CAN contibute to the coin corrosion even though they are not in direct contact. What is important for you is that you change your storage method and location. If this started as a perfectly clean proof when you stored it, you know there's a problem. Time to up your game to save your other coins! I'd probably start with a big container of acetone to fully rinse/dry each coin. Then new holders, coin tubes, plastic bags, desiccant packs and some tupperware.
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