The asymptotic-velocity conjecture for metric lines in Carnot groups

Let $\mathbb{G}$ be a Carnot group with a left-invariant sub-Riemannian structure, Lie algebra $\mathfrak{g}$, and left translations $L_g$. A metric line is a globally minimizing sub-Riemannian geodesic. For a geodesic $\gamma$, its left-translated velocity is $(L_{\gamma^{-1}(t)})_*\dot{\gamma}(t)$.

Asymptotic-velocity conjecture. The metric lines in $\mathbb{G}$ are precisely the sub-Riemannian geodesics $\gamma(t)$ parameterized by arc length for which there exists a unit $v\in\mathfrak{g}$ such that

$$v=\lim_{t\to-\infty}(L_{\gamma^{-1}(t)})_*\dot{\gamma}(t)=\lim_{t\to\infty}(L_{\gamma^{-1}(t)})_*\dot{\gamma}(t).$$

Line geodesics satisfy this condition, and the paper observes that every homoclinic geodesic does as well. The condition does not hold for every heteroclinic geodesic, so the conjecture remains unresolved in general.