The stability conjecture for the infinity-Laplacian and H-type deviation

Let $\mathbb{G}$ be a step two Carnot group with horizontal metric $g_h$. Denote by $\Sigma$ the horizontal layer in exponential coordinates,

$$\Sigma = \{\mathord{\tt g}=\exp({\mathbf x}+{\mathbf t})\in\mathbb{G}:{\mathbf t}=0\}.$$

Let $u$ be the fundamental solution for the sub-Laplacian $\mathcal L$, let $Q$ be the homogeneous dimension, and set $N=u^{1/(2-Q)}$. Here $\delta(\mathbb{G})$ denotes the H-type deviation of $\mathbb{G}$, and $\mathcal L_\infty$ is the horizontal infinity-Laplacian. The stability conjecture.

$$\sup_{\mathord{\tt g}\in\Sigma}\left|N(\mathord{\tt g})(\mathcal L_\infty N)(\mathord{\tt g})\right|\simeq \delta(\mathbb{G})^2.$$

This conjecture proposes that the failure of the fundamental solution's gauge to be infinity-harmonic quantitatively measures the deviation of the group from H-type. The source gives no resolution, so the conjecture remains open.