The local limit conjecture for stable commutator length

Let $G$ be any finitely generated group, let $S$ be a finite symmetric generating set, and suppose that the space $Q(G)$ of homogeneous quasimorphisms modulo homomorphisms is finite dimensional and nonzero. Let $g_n$ be obtained by a length-$n$ random walk with respect to the uniform measure on $S$, conditioned to lie in $[G,G]$. Local limit conjecture. For every $\epsilon>0$, there are positive constants $a,b$, depending on $\epsilon$, such that

$${\mathbf P}\left(a<\frac{\operatorname{scl}(g_n)}{\sqrt n}<b\right)\ge 1-\epsilon$$

for $n\gg0$. This conjecture removes the finite-dimensional homology restriction from the preceding result and predicts square-root-scale stable commutator length under the stated hypotheses.