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 $\partial \mathbb{B}^n$. 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 $n\ge 2$.