The aging conjecture for the running maximum of the Mott random walk

Let $\rho<1$, let $\beta,\lambda\geq0$, and let $Z^{\beta,\lambda}$ be the limiting process. For $h>1$, define

$$\theta(h):=\mathbb{P}\left(\sup_{s\leq1}Z^{\beta,\lambda}_s=\sup_{s\leq h}Z^{\beta,\lambda}_s\right).$$

Aging conjecture.

$$\lim_{n\to\infty}\mathbb{P}^{\beta,\lambda/n}\left(\sup_{s\leq n^{1+1/\rho}}X_s=\sup_{s\leq n^{1+1/\rho}h}X_s\right)=\theta(h)\qquad\text{for every }h>1,$$

with

$$\lim_{h\to1}\theta(h)=1,\qquad \lim_{h\to\infty}\theta(h)=0.$$

This predicts aging of the running maximum: on the anomalous time scale, the maximum over a time interval remains unchanged when the observation interval is enlarged by a factor $h$ with probability tending to $\theta(h)$. The source presents this as an expected property and does not establish it.