Official rule text
33.7 When a player becomes eliminated, their secret objectives 34.4 Abilities, including some found on technology cards, may are shuffled back into the secret objective deck whether those instruct a player to exhaust a card to resolve those abilities. If a secret objectives have been completed or not. card is already exhausted, it cannot be exhausted again.
View official source
Source: Twilight Imperium: Fourth Edition - Living Rules Reference: Prophecy of Kings
Expansion: Prophecy of Kings
Version: 2.0
Reference: Rule 33.7