Asymptotic logarithmic periodicity of the Hungarian roulette probabilities

Let $p_n$ denote the probability associated with the Hungarian roulette model for $n$ players. The preceding concentration estimate indicates that $p_n$ is essentially a weighted mean of $o(n)$ probabilities $p_k$ near $p_{n/e}$.

Asymptotic logarithmic periodicity conjecture. The sequence $p_n$ does not converge as $n\to\infty$ over the positive integers, but it does along every subsequence $(n_j)$ for which the fractional part of $\log n_j$ converges.

This predicts logarithmic periodicity in the asymptotic behavior of the roulette probabilities, while allowing convergence along suitably selected subsequences. The source presents this as a likely behavior rather than establishing it.