Generation of metric scalable groups by the first layer

Let $G$ be a separable nilpotent metric scalable group with geodesic distance, and let $V_1(G)$ denote its first layer.

First-layer generation conjecture. The first layer $V_1(G)$ generates $G$.

For simply connected nilpotent Lie groups admitting dilations, geodicity leads to this generation property because rectifiable curves can be approximated by horizontal line segments. The conjecture asks whether the same conclusion holds for all separable nilpotent metric scalable groups with geodesic distance.