Gromov's Hölder embedding conjecture for the Heisenberg group

Let $\Omega$ be an open subset of $\mathbb{R}^k$, let $n$ be a positive integer, and let $C^{\gamma}(\Omega,\mathbb{H}_n)$ denote the maps that are $\gamma$-Hölder with respect to the Heisenberg metric. An embedding is an injective map that is a homeomorphism onto its image. Gromov's conjecture. There is no embedding

$$f\in C^{\gamma}(\Omega,\mathbb{H}_n)$$

whenever $k\geq n+1$ and $\gamma>\frac{1}{2}$. This conjecture predicts a dimensional obstruction to Hölder embeddings of Euclidean open sets into the Heisenberg group; the cited theorem establishes related non-embedding results under additional Euclidean Hölder regularity assumptions, but the stated conjecture remains open.