Mild random-walk conjecture for one-factorizations

Let $K_n$ be the complete graph on $n$ vertices, and let $\mathcal{G}_n$ be the graph whose vertices are all colorings of $E(K_n)$ with colors ${1,\ldots,n-1}$, with two colorings adjacent when they differ on exactly one edge. A one-factorization is a proper edge-coloring of $K_n$ with $n-1$ colors. In the mild random walk, from a coloring $C$ one moves uniformly to a neighboring coloring $C'$ satisfying $\Psi(C')\leq\Psi(C)$, where $\Psi$ counts pairs of incident equally colored edges. Mild random-walk conjecture. The mild random walk on $\mathcal{G}_n$ started from a uniformly random starting point asymptotically almost surely reaches a one-factorization in $O(n^4)$ steps. The conjecture is presented as supported by numerical evidence; no resolution is given in the source.