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Jon's ability applies a lasting effect on a set of characters at the time you trigger it, in the same way as Spearmaiden. When you satisfy claim, you must choose among those characters. If the claim value is higher than the number of characters that "must be chosen for claim", you must choose all of them, plus additional characters up to the total claim value.
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A character that cannot be killed or leave play (for instance, due to The Crone) cannot be chosen for claim even if it "must be chosen for claim".
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If a character stands or kneels after Jon's ability is used but before resolving claim (for instance, Mya Stone kneeling for her ability and getting canceled), it does not retroactively change whether that character "must be chosen for claim". It is its standing/kneeling status when Jon's ability was triggered that matters.
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Jon's ability triggers just before the C step of the D.U.C.K. sequence. In other words, it triggers after reactions to winning the challenge (for instance, Battle of the Camps), but before intimidate.
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Even though Jon's ability shares the interrupt window with claim replacement abilities, it is not itself a claim replacement ability. It therefore functions alongside such abilities as much as it can, based on the language of the ability in question. For example, if Jon chooses the "standing" option and Vengeance for Elia is played, standing characters must still be chosen for claim even when resolving Vengeance.
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Jon only works when claim is applied "for a challenge". He cannot interrupt cards that work outside of challenges, such as Castamere, Trial by Combat, or Maester Aemon (WotW).
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Even if the claim value is 0, or if there are no characters that can be killed, claim is still applied.
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You can trigger Jon even if you are not the attacking player. In a Melee game, you can trigger him during any challenge, between any two players.