Random stable commutator length conjecture for hyperbolic groups

Let $G$ be a hyperbolic group with finite generating set $S$, and let $\lambda$ be such that the number of elements of length $n$ is $\Theta(\lambda^n)$. Let $v$ be a random element of word length $n$, conditioned to lie in the commutator subgroup $[G,G]$. Random stable commutator length conjecture. For any $\epsilon>0$ and $C>1$,

$$\left|\operatorname{scl}(v)\log(n)/n - \log(\lambda)/6\right| \le \epsilon$$

with probability $1-O(n^{-C})$. The conjecture predicts the typical asymptotic stable commutator length of random homologically trivial elements in arbitrary hyperbolic groups; the source presents it as a technically involved generalization of the results established earlier in the paper, and gives no resolution.