Uniform-entropy cutoff conjecture for random walks with weighted random matching

Let $(G_n)$ be a sequence of graphs with uniformly bounded degrees and diverging sizes. Let $(\varepsilon_n)$ be a sequence of constants in $(0,1)$, and let $(G_n^*)$ be as in the paper's definition of the weighted random matching construction. For a starting vertex $\rho$, let $h_n(\varepsilon_n,\rho)$ denote the entropy of the first long-range edge crossed by the walk on $G_n^*$, and let $h_n(\varepsilon_n)$ denote its average over starting vertices. Uniform-entropy cutoff conjecture. If

$$h_n(\varepsilon_n,\rho_n)\asymp h_n(\varepsilon_n)\quad\text{for all }n\text{ and all }\rho_n,$$

and

$$h_n(\varepsilon_n)\ll\log|V_n|,$$

then the random walk on $(G_n^*)$ exhibits cutoff with high probability, with mixing time of order

$$\frac{\log|V_n|}{\varepsilon_n h_n(\varepsilon_n)}.$$

The conjecture extends the proved results beyond the settings with polynomial ball growth or linear entropy growth. The paper notes that vertex-transitivity automatically gives the required comparability of the entropies, but the conjecture remains unresolved in the supplied text.