Metric-line criterion for normal trajectories in metabelian Carnot groups

Let $G$ be a metabelian Carnot group, let $A\triangleleft G$ be an abelian subgroup with $[G,G]\subseteq A$, and let $V_1$ denote the first layer of $\operatorname{Lie}(G)$. Assume that $(\operatorname{Lie}(A)\cap V_1)^\perp$ is an abelian subalgebra of $\operatorname{Lie}(G)$. Let $\lambda:\mathbb{R}\to T^*G$ be a normal extremal with momentum $\mu\in\operatorname{Lie}(A)^*$, and define $x:=\pi\circ T^*\overline{\Phi}\circ\Pi_\mu(\lambda)$, together with $V_\mu$ and $F_{i,\mu}$, using the notation of the preceding construction. Metric-line conjecture. In this setting, the normal trajectory associated to $\lambda$ is a metric line if and only if, for every $i\in\{1,\ldots,n_1\}$, the limits

$$\lim_{t\to-\infty}F_{i,\mu}(x(t))\quad\text{and}\quad\lim_{t\to+\infty}F_{i,\mu}(x(t))$$

exist and are equal. The preceding corollary rules out metric lines when one of these limits fails to exist or when the two limits differ; the conjecture asserts that this necessary condition is also sufficient, identifying the remaining normal trajectories as the natural candidates for metric lines.