Rigidity conjecture for constant-horizontal-normal sets in Carnot groups

Let $\mathbb G$ be a Carnot group, let $E\subset\mathbb G$ be a set with constant horizontal normal $\nu\in\mathbf S^{m-1}$, and let $H$ be a vertical halfspace with the same horizontal normal. Write $\operatorname{vol}_{\mathbb G}$ for Haar measure on $\mathbb G$, $e$ for the identity, $B_R(e)$ for the metric ball of radius $R$ centered at $e$, and $E\Delta H$ for the symmetric difference. Rigidity conjecture. If

$$\liminf_{R\to+\infty}\frac{\operatorname{vol}_{\mathbb G}\bigl((E\Delta H)\cap B_R(e)\bigr)}{\operatorname{vol}_{\mathbb G}\bigl(B_R(e)\bigr)}=0,$$

then $E$ is a vertical halfspace.

This conjecture would provide the monotonicity or stability principle needed to complete the rectifiability theory for sets of finite perimeter in Carnot groups. The surrounding result proves that vertical halfspaces occur as tangents almost everywhere, while the conjecture asserts the rigidity of constant-horizontal-normal sets that become asymptotically indistinguishable from a vertical halfspace along a sequence of scales.