Relevant text:
Bayushi Kan XP Elemental Battle/Open: Target your Personality you control and target another player: After this action ends, the target player is considered to have targeted your Personality with an action.
Hida Osote After the first time each turn another player's action targets your Crab Clan Personality at Osote's location: The player discards a card after the action ends.
A player controls both Kan XP and Osote. He uses Kan's ability, choosing Osote. The opponent is "considered to have targeted Osote with an action", which would appear to satisfy the conditions for Osote's trait to trigger. However, since there isn't actually an action, that action never "ends" (or does it end immediately?). Does the opponent have to discard a card to Osote's effect?