If Hotaru / Toturi win a conflict at Pilgrimage, but do not break it, do they get to resolve the ring effect once from their action?

Gut tells me "no", but it isn't entirely clear from the cards.

You claimed the ring, and the reaction is to claiming, not to it firing so I would have to say yes. You get one use instead of two.

You don't. Pilgrimage cancels all ring effects, and Hotaru/Toturi have you resolve the ring's effect.

Pilgrimage literally says " cancel all ring effects "