I'd just leave it alone. You can "prune" the dead skeletons if they impede the flow through the colony, but in your case that doesn't look like an issue. You can do either though.
Nothing will "come out" of the dead stalk. The live part of the coral is the tissue that's wrapped around the end of the stalk. As the coral grows, it will produce more skeleton (stalk) under the tissue, and the tissue will grow up the stalk. The "border" of the living tissue may move up the stalk, or not. As it grows, the stalks will get taller, and will often branch off forming multiple shoots where there used to be one. If something happens to a head, or you break/cut/frag it, you'll be left with the dead stalk (skeleton).
Sorry if I'm repeating what you already know, but your comment about it "coming out" reminded me that I had no clue how these things grew when I got my torch!