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It seems that after defeating the King of the Dead with his companion, you just need to chat a bit. As for the station, there are different options, as I understand it. In my case, I had to repair a fuse in one relay and remove a conductive element in the other.
I replayed the dungeon with a different build and defeated the curse without any problems

, and even improved the result by 8 days. +18 Phoenixes from Gluttony is a bit too much, especially considering everything else.