I was re-reading it just now and it dawned on me all of a sudden (pardon if this was brought up before or it happens that my interpretation is wrong):
You can win a challenge as a defender and you must still satisfy the claim along with all other opponents (including the attacker).
I think the card could have been better phrased:
"After you win a challenge as an attacker in which you control Melee Champion as a participant, each opponent (in addition to the defending player) must also fulfill the claim of that challenge."
ALSO
"House X Only"
where House X is the house that won the pertinent storyline league.