Regularity of subFinsler spheres in Carnot groups

Let $G_\infty$ be the asymptotic cone, equipped with its Pansu limit metric $d_\infty$, and let a Riemannian distance on $G_\infty$ be fixed. The unit sphere is the boundary of

$$B_{d_\infty}(\operatorname{id},1)=\{g\in G_\infty:d_\infty(\operatorname{id},g)\leqslant 1\}.$$

Regularity conjecture. The unit sphere of $d_\infty$ is rectifiable with respect to every Riemannian distance on $G_\infty$. In particular, if $G_\infty$ has topological dimension $n$, then the sphere has finite $(n-1)$-dimensional Lebesgue measure. This remains open for general Carnot groups and is proposed because such regularity would imply the desired volume asymptotics for the associated Stoll metric.