Moment equivalence conjecture for stable-process passage times and functionals

Let $Z=(Z_t)_{t\geq 0}$ be the stable process considered in the paper, let $T$ be the associated stopping time, and let $\alpha$ be its stability index. For every $k>0$, the moments of $T$ and $Z_T$ satisfy

$$\mathbb{E}[T^k]<+\infty\quad\Longleftrightarrow\quad\mathbb{E}[Z_T^{k\alpha}]<+\infty.$$

This connects the lower-tail behavior of the homogeneous functional $Z_T$ with that of the stopping time $T$; the source presents it as a statement to be raised from the preceding inequalities and does not provide evidence of its resolution.