  Let $M$ be an open (complete and non-compact) manifold with $\mathrm{Ric}\ge
0$ and escape rate not $1/2$. It is known that under these conditions, the
fundamental group $\pi_1(M)$ has a finitely generated torsion-free nilpotent
subgroup $\mathcal{N}$ of finite index, as long as $\pi_1(M)$ is an infinite
group. We show that the nilpotency step of $\mathcal{N}$ must be reflected in
the asymptotic geometry of the universal cover $\widetilde{M}$, in terms of the
Hausdorff dimension of an isometric $\mathbb{R}$-orbit: there exist an
asymptotic cone $(Y,y)$ of $\widetilde{M}$ and a closed $\mathbb{R}$-subgroup
$L$ of the isometry group of $Y$ such that its orbit $Ly$ has Hausdorff
dimension at least the nilpotency step of $\mathcal{N}$. This resolves a
question raised by Wei and the author, and extends previous results on virtual
abelianness by the author.
