I would try two things.

1. heat gun the goo a little (get it up to like 150F or so) and see if it the sappy liquid runs and you can maybe clean it and drain it easier at temp 2)get some shrink wrap to seal up the ends. That might help for a while if you can get it tight.

I’m not familiar with those headphones but sometimes manufacturers will put grease or goo inside the cables to reduce straining and folding of the wires.

It looks kinda like rosin in the photo. I wonder if something inside the device was saturated in rosin and its just leaking down the cable?