The finite-dimensionality conjecture for quasimorphisms

Let $G$ be a finitely presented group, and let $g_n$ be obtained by a length-$n$ random walk, conditioned to lie in $[G,G]$. Suppose that, for every $\epsilon>0$, there is a positive constant $b$ such that

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

for $n\gg0$. Finite-dimensionality conjecture. Then $Q(G)$ is finite dimensional. Here $Q(G)$ denotes the space of homogeneous quasimorphisms modulo homomorphisms. The conjecture proposes that an $O(\sqrt n)$ upper bound for random stable commutator length characterizes finite-dimensionality of $Q(G)$ among finitely presented groups; the source supplies no resolution.