Stoll's bounded-distance conjecture for word and subFinsler metrics

Let $G$ be a simply connected nilpotent Lie group, let $\Gamma$ be a discrete co-compact subgroup, and let $S$ be a finite symmetric generating set containing the identity. Let $\rho_S$ be the word metric on $\Gamma$, let $d_S$ be the associated Stoll metric, and let $B_S(n)=S^n$. If the Stoll metric is at bounded distance from the word metric, then the ball-volume asymptotics can be reduced to the corresponding asymptotics for $d_S$. Stoll metric conjecture. The bounded-distance approximation asserted above should hold, and consequently one expects

$$|B_S(n)|=c_S n^d+O_S(n^{d-1})$$

for all finitely generated nilpotent groups. The source says that this remains conjectural in higher nilpotency step and links it to the volume-error conjecture.