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So the boiling method worked. I stuck the collar in a pot, covered with water, added a dash of liquid detergent, and let it bubble for about 15 minutes.
The gum disappeared entirely, disolving, I guess. And the pan survived unscathed as well. Also, the cleanest part of the collar is now the little circle where the gum was. It's a bright and cheery spot, while the rest is a comfortable, lived-in shade of green. Guess we discovered a whole new way to do laundry!