Polynomial-time convergence conjecture for non-increasing edge recoloring

Let $K_n$ be the complete graph on $n$ vertices, and let $\Phi$ be the potential function on colorings of $E(K_n)$ with $n-1$ colors. Consider the process that starts from a random coloring, repeatedly chooses a random edge $e$ and a random color $c$, and recolors $e$ to $c$ if this does not increase $\Phi$. Polynomial-time convergence conjecture. With probability $1-o_n(1)$, this process converges to a one-factorization of $K_n$ in time $\operatorname{polynomial}(n)$. The source presents this as a belief based on simulations and does not establish the claimed high-probability polynomial-time convergence.