Log-normal limit conjecture for increasing Hamiltonian paths

For each $n$, let $X$ be the number of Hamiltonian paths that are increasing in a uniformly random ordering of the edges of $K_n$. Log-normal limit conjecture. As $n\to\infty$, the normalized variable $X/n$ converges in distribution to a $\log \mathcal{N}(-1/2,1)$ random variable. The existence and identification of a limiting distribution for $X_n/n$ remain open in the supplied text; the preceding theorem gives evidence for this conjecture through the limiting properties of all subsequential distributions.