What to do with the Broken Relic in Cuphead

It's hard to know what to do with the Cuphead Broken Relic. It doesn't give you any info in the description, and the only clue you get is from that weird detective ghost, who hints that it's linked with the climbing competition near the mountain and the graveyard. If you manage to solve the mystery, you'll unlock a secret boss and an extra-challenging game mode...
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