The limsup law for normalized long-range dependent processes

Let $p\geq 1$, and let $Y_{n,p}$ and $\sigma_{n,p}$ denote the process and normalizing sequence defined in the paper. Limsup conjecture.

$$\limsup_{n\to\infty}\sigma_{n,p}^{-1}(\log\log n)^{-p/2}Y_{n,p}=c(\beta,p)\quad\text{almost surely}.$$

This conjecture would identify the optimal iterated-logarithm rate for the processes considered and, in particular, would support removing the factor $(\log n)^{1/2}$ from the relevant uniform bound at least when $\beta<2/3$.