I finally did the same thing @mrmuro describes in his comment – except I used a knfe as a cantilever to pry a bit to loosen the glue inside before I twisted it off. It's stuck pretty good.
These are the comments accompanying the pictures in the link:
http://imgur.com/a/UAyGt
"It's hard to see what's going on there due to the clap and tape, and all the glue underneath.
[I removed the] clamp, tape and [pushed] the cable in to make room
in the circle you can see the wire that has gotten caught by the cable when it twists inside the joint. The result of getting caught is that it is tugged, and thus reducing the contact with the soldering point underneath the glue.
I forgot to take a picture of what I did afterwards, but I removed the glue and soldered the blue cable down onto the board again.
when reassembling, try to place the tiny cables in a way so they will not get caught by the twisting black rubber cable in the joint."
Disclaimer: You may have another tiny cable at fault, but in my case, it was the blue one.
Good luck!