Stoll metric bounded-distance conjecture

Let $G$ be a simply connected nilpotent Lie group and let $\Gamma$ be a discrete co-compact subgroup. For a finite symmetric generating set $S\subset\Gamma$, let $\rho_S$ be the associated left-invariant word metric. Identify $G$ with its Lie algebra by the exponential map, let $V_S$ be the span of $S$, and let $d_S$ be the left-invariant subFinsler metric whose norm on $V_S$ has unit ball equal to the convex hull of $S$. This is the Stoll metric associated with $(\Gamma,S)$. Stoll metric conjecture. There is a constant $C=C(S)>0$ such that

$$|\rho_S(\operatorname{id},\gamma)-d_S(\operatorname{id},\gamma)|\leqslant C$$

for all $\gamma\in\Gamma$. The source presents this as a candidate metric approximation and does not provide a proof; it is used later to reduce the volume conjecture to a continuous asymptotic problem.