Consensus propagation mixing-time conjecture for the m-dimensional torus

Let the graph be the $m$-dimensional torus with $n$ nodes, and let $\tau^\star$ denote the mixing time of the directed-edge process associated with consensus propagation. Consensus propagation mixing-time conjecture. For the $m$-dimensional torus with $n$ nodes,

$$\tau^\star = \Theta(n^{(2m-1)/m^2}).$$

For comparison, the optimized vertex-process mixing time satisfies $\tau_2=\Theta(n^{2/m})$; the conjecture predicts the mixing time of consensus propagation and would clarify its advantage over pairwise averaging on higher-dimensional grids. The source presents this as a conjecture without resolving it.