Okay, I have a little more information.
When I tried to turn in the pieces from the cave and the southeast slope close to Perdition Hold, they glitched.
While standing in the circle, I joined a cross-server group for the quest. One of the two pieces I had (the one from the cave) disappeared, and I got credit for it. The other piece also vanished, but I didn’t get credit.
I ran out of the circle but couldn’t see the piece that had vanished. Even after I dropped group and reverted back to my original server, I couldn’t see the piece. I tried Death Gating to Acherus and back, but that didn’t fix it.
I logged out. That fixed it, at least enough for me to see the piece that had glitched. It was the one close to where the turn-in is, on the slope southeast of Perdition Hold
I dragged the piece to the circle, joined the original group, and lost the third piece again.
I dropped group, logged out again, dragged the piece to the circle, and joined a group on an entirely different server.
This time, the quest completed and I was able to progress.