Digraph analogue of Cereceda's diameter conjecture

Let $D$ be a digraph on $n$ vertices, and let $\delta^*_{\min}(D)$ denote its min-degeneracy. Let ${\cal D}_k(D)$ be the graph whose vertices are the $k$-dicolourings of $D$, with adjacency given by recolouring one vertex. Digraph Cereceda conjecture. If $k\geq \delta^*_{\min}(D)+2$, then the diameter of ${\cal D}_k(D)$ is at most $O(n^2)$. This conjecture extends Cereceda's quadratic-diameter conjecture from graphs to digraphs; the paper proves linear-diameter results for larger colour ranges, but the stated quadratic bound remains open.