  Suppose the edges of the complete graph on $n$ vertices are assigned a
uniformly chosen random ordering. Let $X$ denote the corresponding number of
Hamiltonian paths that are increasing in this ordering. It was shown in a
recent paper by Lavrov and Loh that this quantity is non-zero with probability
at least $1/e-o(1)$, and conjectured that $X$ is asymptotically almost surely
non-zero. In this paper, we prove their conjecture. We further prove a partial
result regarding the limiting behaviour of $X$, suggesting that $X/n$ is
log-normal in the limit as $n\rightarrow\infty$. A key idea of our proof is to
show a certain relation between $X$ and its size-biased distribution. This
relies heavily on estimates for the third moment of $X$.
