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Bertoin and Yor proved that $X_\\xi$ is moment-determinate when $\\xi$ has no positive jumps, and conjectured that this condition is also necessary. We prove the latter.\n  The proof is based on a lower bound near zero for the law of $I_\\xi$. We show that a group of sufficiently many positive jumps near the origin puts $I_\\xi$ on a suitable small scale. 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