I know you'll poo-poo this idea, I know I would if I didn't know any differently. But a highly acclaimed carpet cleaner told me that the best thing was BBQ lighting gel. My first ever school job we actually resorted to using it with fantastic, though smelly (Paraffiny) results. The headmaster was well chuffed with the results. The carpets came to no harm as we clean them every year...
Apparently chewing gum is a petroleum derivative which is why it emulsifies when it comes into contact with the lighter gel....