The largest Carnot subgroup conjecture for Markov convexity

Let $G$ be a graded nilpotent Lie group, and let $H$ be its largest Carnot subgroup. For $p>0$, write $G$ as Markov $p$-convex when its metric space satisfies the Markov $p$-convexity inequality.

Largest Carnot subgroup conjecture. The set of $p$ for which $G$ is Markov $p$-convex is the same as the set of $p$ for which $H$ is Markov $p$-convex.

This conjecture proposes that Markov convexity of a graded nilpotent Lie group is completely determined by its largest Carnot subgroup. The source presents it as an additional conjecture, with no resolution stated.