The integrated stable-process first-passage tail conjecture

Let $Z$ be a Lévy stable process, let $\rho$ be its positivity parameter, and suppose that $|Z|$ is not a subordinator. Define

$$A_t=\int_0^t Z_s\,\mathrm{d}s,$$

with first-passage time

$$T=\inf\{t>0:A_t=1\}.$$

Integrated stable-process tail conjecture. One has

$$\mathbb{P}[T>t]=t^{-\rho/2+o(1)},\qquad t\to+\infty.$$

This conjecture is motivated by the corresponding ruin-probability asymptotics for $Z$. The source marks it as resolved: the case with no negative jumps, $\alpha>1$ and $\rho=(\alpha-1)/\alpha$, was proved in the cited work, with sufficient control of the error term to imply the stated moment divergence.