  It is unknown if there exists a locally $\alpha$-H\"older homeomorphism
$f:\mathbb{R}^3\to \mathbb{H}^1$ for any $\frac{1}{2}< \alpha\le \frac{2}{3}$,
although the identity map $\mathbb{R}^3\to \mathbb{H}^1$ is locally
$\frac{1}{2}$-H\"older. More generally, Gromov asked: Given $k$ and a Carnot
group $G$, for which $\alpha$ does there exist a locally $\alpha$-H\"older
homeomorphism $f:\mathbb{R}^k\to G$? Here, we equip a Carnot group $G$ with the
Carnot-Carath\'eodory metric. In 2014, Balogh, Hajlasz, and Wildrick considered
a variant of this problem. These authors proved that if $k>n$, there does not
exist an injective, $(\frac{1}{2}+)$-H\"older mapping $f:\mathbb{R}^k\to
\mathbb{H}^n$ that is also locally Lipschitz as a mapping into
$\mathbb{R}^{2n+1}$. For their proof, they use the fact that $\mathbb{H}^n$ is
purely $k$-unrectifiable for $k>n$. In this paper, we will extend their result
from the Heisenberg group to model filiform groups and Carnot groups of step at
most three. We will now require that the Carnot group is purely
$k$-unrectifiable. The main key to our proof will be showing that
$(\frac{1}{2}+)$-H\"older maps $f:\mathbb{R}^k\to G$ that are locally Lipschitz
into Euclidean space, are weakly contact. Proving weak contactness in these two
settings requires understanding the relationship between the algebraic and
metric structures of the Carnot group. We will use coordinates of the first and
second kind for Carnot groups.
