BiLipschitz embeddings of spheres into jet space Carnot groups not admitting Lipschitz extensions

For all $k,n\ge 1$, we construct a biLipschitz embedding of ${\mathbb{S}}^n$ into the jet space Carnot group $J^k({\mathbb{R}}^n)$ that does not admit a Lipschitz extension to ${\mathbb{B}}^{n+1}$. Let $f:{\mathbb{B}}^n\to \mathbb{R}$ be a smooth, positive function with $k^{th}$-order derivatives that are approximately linear near $\mathrm{\partial }{\mathbb{B}}^n$. The embedding is given by taking the jet of $f$ on the upper hemisphere and the jet of $-f$ on the lower hemisphere, where we view ${\mathbb{S}}^n$ as two copies of ${\mathbb{B}}^n$. To prove the lack of a Lipschitz extension, we apply a factorization result of Wenger and Young for $n=1$ and modify an argument of Rigot and Wenger for $n\ge 2$.