Lavrov–Loh conjecture on Hamiltonian monotone paths in random edge-orderings

Let $K_n$ be the complete graph on $n$ vertices, choose an edge-ordering of $K_n$ uniformly at random, and call a path Hamiltonian if it contains all vertices of $K_n$. A path is monotone when it traverses edges in increasing order in the edge-ordering.

Lavrov–Loh conjecture. With probability tending to $1$ as $n\to\infty$, a random edge-ordering of $K_n$ contains a Hamiltonian monotone path.

The conjecture strengthens the result that the probability is at least $1/e-o(1)$. It asks whether Hamiltonian monotone paths occur with asymptotically overwhelming probability; the source gives no resolution.