The concentration conjecture for random stable commutator length

Let $(G,\mu,Y)$ be nondegenerate in the sense of Definition~, and let $g$ be obtained by an even-length random walk on $G$, conditioned to lie in $[G,G]$. Concentration conjecture. There are constants $C>0$ and $C_1>0$ such that, for every $\epsilon>0$,

$${\mathbf P}\left(\left|\operatorname{scl}(g)\frac{\log n}{n}-C\right|\le\epsilon\right)\ge 1-e^{-n^{C_1}}.$$

This is proposed as a sharpening of the paper's order-of-growth estimates for stable commutator length; no resolution is supplied.