Interior-point conjecture for wedges generated by horizontal half-spaces

Let ${\mathbb G}$ be a Carnot group and let $W$ be a horizontal half-space. Denote by $S_W$ the semigroup generated by $W$, by $\bar S_W$ its closure, by $\mathfrak w_{\bar S_W}$ the wedge tangent to $\bar S_W$, and by $\operatorname{Int}(S_W)$ the interior of $S_W$. Interior-point conjecture. For every horizontal half-space $W$ in every Carnot group ${\mathbb G}$,

$$\exp(\mathfrak w_{\bar S_W})\cap \operatorname{Int}(S_W)\neq\emptyset.$$

There is evidence that the intersection is nonempty in Carnot groups of step at most $4$; whether this holds in general remains open.