Second-order limit conjecture for the renewal covering

Let $(P_n)_n$ be a renewal covering of the natural numbers as defined in the paper. Let $Z$ be a random variable. Second-order limit conjecture.

$$\frac{P_n-n\mathrm{ln}(n)-n\ln\mathrm{ln}(n)}{n}\overset{D}{\implies}Z,$$

where the convergence is in distribution.

The paper proves the first-order behavior $P_n\sim n\mathrm{ln}(n)$ almost surely and expects the displayed normalization to capture a second-order term, while stating that the corresponding third-order behavior does not persist. The conjecture is left open.