The inverse-distance Lipschitz conjecture for sub-Finsler Carnot groups

Let $\mathbb G$ be a sub-Finsler Carnot group of step $2$, with Carnot–Carathéodory distance $d_{\mathrm{cc}}$, abnormal set $\mathrm{Abn}$, and a fixed Riemannian metric $d_{\mathrm{eu}}$. Let $K\subset\mathbb G$ be compact with $0_{\mathbb G}\notin K$. The inverse-distance Lipschitz conjecture. There exists a constant $C>0$ such that, for every $\delta>0$, the function $d_{\mathrm{cc}}(0_{\mathbb G},\cdot)$ is $C\delta^{-1}$-Lipschitz with respect to $d_{\mathrm{eu}}$ on

$$\{g\in K:d_{\mathrm{eu}}(g,\mathrm{Abn})\geq\delta\}.$$

The conjecture would, together with the cited Sard-property result and the arguments of the paper, imply Euclidean codimension-one rectifiability of sub-Finsler spheres for all sub-Finsler Carnot groups of step $2$. The stated inverse-distance bound is known to be optimal in an example, but no faster divergence is currently known.