The Markov convexity threshold conjecture for Carnot groups

A Carnot group is a graded nilpotent Lie group generated by its first stratum. For $p>0$, a metric space is Markov $p$-convex when it satisfies the Markov $p$-convexity inequality. Let $G$ be a Carnot group of step $r$.

Markov convexity threshold conjecture. $G$ is not Markov $p$-convex for every

$$p\in(0,2r).$$

The preceding results establish Markov $p$-convexity for Carnot groups in the complementary range relevant to the conjecture, so the claim would identify the sharp threshold at $2r$. Its general validity is left open in the source.