The entropic-time conjecture for $L_2$ cutoff on random Cayley graphs

Let $G$ be a finite group and let $G_k$ be its random Cayley graph. For ${L}\in\mathbb{Z}\cup\{\infty\}$, let $\mathbb{Z}_{L}$ denote the cyclic group of order $L$, with $\mathbb{Z}_{\infty}=\mathbb{Z}$, and let $\tilde t_L^\pm(k,G)$ be the time $t$ at which the return probability of the random walk on $\mathbb{Z}_{L}^k$ at time $2t$ equals $|G|^{-1}$. Set

$$\tilde t_*^\pm(k,G)=\max_{L\in\mathbb{N}}\tilde t_L^\pm(k,G).$$

The entropic-time conjecture. Under conditions similar to those in the paper's total-variation cutoff result, with high probability the random walk on $G_k$ exhibits cutoff in the $L_2$ metric at time $\tilde t_*^\pm(k,G)$. This would identify the $L_2$ cutoff scale through return probabilities on the comparison walks on $\mathbb{Z}_L^k$. The statement is presented informally and the precise conditions are deferred to the cited total-variation result; the paper proves related refined comparisons but does not establish this conjecture in full.