Failure of the curvature-dimension condition in sub-Finsler Carnot groups

Let $G$ be a sub-Finsler Carnot group, endowed with a positive smooth measure $\mathfrak m$. Let $\mathsf d_{SF}$ denote its sub-Finsler distance, and let $\mathsf{CD}(K,N)$ be the curvature-dimension condition for metric measure spaces.

Failure of the curvature-dimension condition conjecture. The metric measure space $(G,\mathsf d_{SF},\mathfrak m)$ does not satisfy the $\mathsf{CD}(K,N)$ condition for any $K\in\mathbb{R}$ and $N\in(1,\infty)$.

This conjecture predicts that no sub-Finsler Carnot group with a positive smooth measure satisfies any finite-dimensional curvature-dimension condition, extending the expected obstruction caused by the singularity of the norm beyond the sub-Riemannian setting. Its resolution is not established in the supplied text.