Generic extremals avoid the strict-subgroup union in the free step-2 rank-4 group

Let $\mathbb F_4$ be the free step-2 rank-four Carnot group, and let $\mathcal H$ be the union of all strict Carnot subgroups. Equivalently,

$$\mathcal H=\{(x,t)\in\mathbb F_4:\operatorname{rank}(t)\leq 2\}.$$

Let $\gamma=\gamma(\cdot,a,b,\varphi)$ be a generic extremal. Generic-extremal avoidance conjecture. The image of $\gamma$ at positive times does not meet $\mathcal H$:

$$\gamma((0,+\infty))\cap\mathcal H=\varnothing.$$

The paper proves only that any such intersection, if it occurs, lies after the cut time and before a positive extremal-dependent constant $c_2(\gamma)$; the conjectured global avoidance remains open.