It says "until the regroup phase," so if the Fellowship begins their turn on that site, its effect should last until the next regroup phase. And there is precedent for sites' text being active even when the fellowship isn't there. See Mount Doom.
Hmm, I've read the card as "minions skirmishing
companions lose fierce and can't gain it again until regroup phase". If you read it "Until regroup phase, if minion skirmishes a
companion it loses fierce and cannot gain it again", then the "lose and can't gain" is for how long? Until end of turn?