It’s not that they can’t die. Hell, Devos was legit tryna kill the Archon, and it’s stated that the Jailer’s goal was to kill the Arbiter. And the Arbiter is also an Eternal One. It’s just that the members respective realms get fucked in the process, unless there is a successor, or a new leader in place/entrusted upon before they die (I think).
The strength of the realms also connect to the strength of the Eternal One. It’s why the Jailer’s power grows infinitely while new souls go to the Maw
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Also, I doubt the Jailer wants to kill the Archon. He probably wants to torture her or fuck with her tbh. Pretty sure torturing the Eternal Ones as payback for what he’s gone through is exactly what the Jailer wants.