The almost-sure existence conjecture for increasing Hamiltonian paths

Let $K_n$ be the complete graph on $n$ vertices, and let $f$ be an edge ordering of $K_n$, chosen uniformly at random. An $f$-increasing Hamiltonian path is a Hamiltonian path whose edges occur in increasing order under $f$.

Increasing Hamiltonian path conjecture. With probability tending to $1$ as $n\to\infty$, $K_n$ contains an $f$-increasing Hamiltonian path.

The preceding theorem establishes only probability at least $1/e+o(1)$, while earlier results give almost-Hamiltonian increasing paths asymptotically almost surely. Numerical simulations suggest that the stronger almost-sure Hamiltonian statement should hold.