Bertoin-Yor Moment Determinacy Conjecture

For an unkilled Levy process $\xi$ drifting to $+\infty$ with all positive exponential moments, let $I_\xi = \int_0^\infty e^{-\xi_t}\,dt$ and $X_\xi = 1/I_\xi$. Bertoin and Yor proved $X_\xi$ is moment-determinate when $\xi$ has no positive jumps and conjectured that this condition is necessary. The conjecture is settled.