In a recent store championship, I did an intrigue challenge which I won by more than 5 STR. My opponent had no cards in hand, and therefore did not have to discard any cards. I played Insidious Scheme. My opponent questioned whether or not I could play Insidious Scheme even though he did not have to do anything to satisfy claim. My belief, and the consensus that others agreed with, was that "After claim is applied" refers to the "Apply Claim" step resolving regardless of the actual claim result.

That said, we could find no explicit ruling on this matter and I thought I would toss it to the community for confirmation.

In a recent store championship, I did an intrigue challenge which I won by more than 5 STR. My opponent had no cards in hand, and therefore did not have to discard any cards. I played Insidious Scheme. My opponent questioned whether or not I could play Insidious Scheme even though he did not have to do anything to satisfy claim. My belief, and the consensus that others agreed with, was that "After claim is applied" refers to the "Apply Claim" step resolving regardless of the actual claim result.

That said, we could find no explicit ruling on this matter and I thought I would toss it to the community for confirmation.

Yup, you were right. The relevant explanation touches on this line, in "Intrigue Claim" of the Rules:

" When a player loses an intrigue ( ) challenge as the defending player, that player must discard a number of random cards from his or her hand equal to the claim value on the attacking player’s revealed plot card. If the player has fewer cards in hand than would satisfy claim, as much of the claim as possible must be satisfied. "

That is to say, the rules allow you to apply claim but discard no cards (in fact, you can apply a claim of 0), with the basic idea that it works as an "if able" situation.

Furthermore, while a player can't initiate an ability that wouldn't change the game state, the game can - forced reactions, when revealeds, claim, dominance and other framework actions are still initiated by the game, and can resolve unsuccessfully. The game doing so still creates a triggering condition.