Metric-line classification conjecture for Carnot groups

Let $\mathbb{G}$ be a Carnot group with left-invariant sub-Riemannian structure. A curve $\gamma(t):\mathbb{R}\to\mathbb{G}$ is a metric line if it is a globally minimizing sub-Riemannian geodesic. Metric-line classification conjecture. The metric lines in $\mathbb{G}$ are precisely the sub-Riemannian geodesics $\gamma(t)$ parameterized by arc length for which there exists a unitary vector $v\in\mathfrak{g}$ such that

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

where $L_g$ denotes left translation by $g\in\mathbb{G}$ and $(L_g)_*$ its push-forward. This conjecture seeks to characterize global minimizers among geodesics in Carnot groups; the paper attacks it for metabelian Carnot groups with semidirect product structure, including Engel-type groups, but the general classification remains open.