Does Discipline persist after a Showdown ends?

Yes, Discipline's +2 Might persists after a showdown ends. The effect says "this turn," and a showdown is just a structured window of opportunity within the turn — it does not truncate "this turn" durations.

Key points:

• Discipline gives +2 Might until the Expiration Step of your Ending Phase (Rule 317.2.d), where all "this turn" effects expire simultaneously. A showdown ending does not override that timing.
• During Combat Cleanup (Resolution Step, Rule 461.1.a), all units heal damage. Your unit keeps the +2 from Discipline through that healing, then the buff continues until the turn ends.
• This is consistent with FAQ #3532: the +2 Might lasts the entire turn regardless of whether the unit took damage or participated in a showdown.

So your unit keeps the boost for the rest of the turn — into additional combats, follow-up moves, or the Ending Phase.