Wilson's extremal mixing conjecture for random Cayley graphs

Let $d$ and $n$ diverge with $n\leq 2^d$, let $G$ be a group of order $n$, and let $H=\mathbb{Z}_2^d$. For fixed $\varepsilon,\varepsilon'\in(0,1)$, write $t_{\mathrm{mix}}(\varepsilon,G_k)$ for the mixing time of the random walk on $G_k$. Wilson's conjecture. If

$$k-\log_2 n\gg 1,\qquad \log k\ll\log n,$$

then, with high probability,

$$\frac{t_{\mathrm{mix}}(\varepsilon,G_k)}{t_{\mathrm{mix}}(\varepsilon',H_k)}\leq 1+o(1).$$

Thus the random walk on $G_k$ is no slower than that on $\mathbb{Z}_2^d{}_k$, up to lower-order terms. The paper proves the corresponding statement for Abelian groups, but the all-groups conjecture remains open.