The persistence exponent conjecture for integrated stable Lévy processes

Let $Z$ be a strictly $\alpha$-stable Lévy process, let $A_t=\int_0^tZ_s\,\mathrm{d}s$, and define $T_1=\inf\{t\ge0:A_t>1\}$. The stable Lévy persistence conjecture. If

$${\mathbb{P}}[Z_1>0]=\rho\in(0,1),$$

then there exists $c>0$ such that

$${\mathbb{P}}[T_1>t]\sim c\,t^{-\rho/2}.$$

The paper notes that this is proved only in the spectrally positive case up to the stated refinement, while the general case remains open.