Star-graph averaging conjecture

Let $S_n$ be the star graph on $n+1$ vertices, and let $x$ be the weight vector on $S_n$ that gives weight $1-\frac{1}{n+1}$ to the central vertex and weight $-\frac{1}{n+1}$ to every other vertex. Define

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

Here $\overline{x}$ denotes the average-weight vector. Star-graph averaging conjecture. For $t=o(n\log n)$,

$$\mathbb{E}[\Delta_1(t,x)]\sim\lVert x-\overline{x}\rVert_2.$$

The conjecture asserts that the $n\log n$ mixing scale suggested by the general bound is essentially tight for this focused initial condition on a star graph. The paper gives numerical evidence but no proof or resolution.