The lower-central-series comparison conjecture for random Cayley graphs

Let $G$ be a group with lower central series $(G_\ell)_{\ell\geq 0}$, and let

$$L=\min\{\ell\geq 0\mid G_\ell=\{\mathsf{id}\}\}.$$

Write the prime decomposition of $|G_L|$ as $|G_L|=\prod_{j=1}^r p_j$, and define

$$\overline G=\left(\bigoplus_{\ell=1}^{L}G_{\ell-1}/G_\ell\right)\oplus\left(\bigoplus_{j=1}^{r}\mathbb{Z}_{p_j}\right).$$

Lower-central-series comparison conjecture. If

$$1\ll\log k\ll\log|G|,\qquad k-d(\overline G)\gg 1,$$

then, with high probability,

$$\frac{t_{\mathrm{mix}}(G_k)}{t_{\mathrm{mix}}(\overline G_k)}\leq 1+o(1).$$

This extends the paper's nilpotent and Abelian comparison results and contains Wilson's conjecture as a special case. The general comparison remains open.