Path-graph averaging conjecture without a logarithmic factor

Let $P_n$ be the path graph on $n$ vertices. For a weight vector $x$ on $P_n$, define

$$\Delta_1(t,x)=\lVert\mathbf{\mathrm{R}}_{P_n}^{t}(x)-\overline{x}\rVert_1,$$

where $\overline{x}$ denotes the average-weight vector. Path-graph averaging conjecture. For every $c>0$, there exists a constant $M=M(c)$ such that, for every weight vector $x$ of $P_n$,

$$\mathbb{E}[\Delta_1(Mn^3,x)]\le e^{-c}\lVert x-\overline{x}\rVert_2.$$

This conjecture says that the logarithmic factor in the general convergence bound is unnecessary for paths. The paper motivates it with numerical experiments, but gives no proof or resolution.