The norm conjecture for random stable commutator length subspaces

Let $(G,\mu,Y)$ be nondegenerate in the sense of Definition~. Let $g_1,\ldots,g_k$ be obtained by independent even-length random walks on $G$ of lengths $\ell_1n,\ldots,\ell_kn$, conditioned to lie in $[G,G]$. Norm conjecture. There are constants $C>0$ and $C_1>0$ such that, for every finite integer $k$, every $\epsilon>0$, and the corresponding formal coefficients $t_i$,

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

The source says this conjecture would follow from the concentration conjecture and presents it as a conjectural description of the scl norm on a random subspace; it is not resolved here.