The persistence exponent conjecture for general integrated Lévy processes

Let $Z$ be a Lévy process, let $A_t=\int_0^tZ_s\,\mathrm{d}s$, and for $x>0$ define $T_x=\inf\{t\ge0:A_t>x\}$. The general Lévy persistence conjecture. If

$${\mathbb{P}}[Z_t>0]\to\rho\in(0,1)\quad\text{as }t\to+\infty,$$

then for every $x>0$,

$${\mathbb{P}}[T_x>t]=t^{-\rho/2+o(1)}.$$

This is posed as a generalization of the discrete and stable-process conjectures; no proof is supplied.