On a moment determinacy conjecture of Bertoin and Yor

Let $\xi$ be an unkilled real-valued Lévy process which drifts to $+\infty$ and has positive exponential moments of all orders, and define $I_\xi=\int_0^\infty e^{-\xi_t},dt$, and its reciprocal $X_\xi=1/I_\xi$. 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.
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. The first selected jump time is used as a one-dimensional smooth coordinate, yielding an absolutely continuous subcomponent of the law of $I_\xi$. After the change of variables, the resulting subdensity of $X_\xi$ satisfies a Krein moment indeterminacy criterion.