The hyperplane asymptotic conjecture for non-geodesic horizontal projections

Let $G$ be a Carnot group with first layer $V_1$, let $\pi:G\to G/[G,G]$ be the quotient map, and let $d$ denote the relevant distance. Suppose that $\gamma:\mathbb{R}\to G$ is a geodesic whose horizontal projection $\pi\circ\gamma:\mathbb{R}\to G/[G,G]$ is not a geodesic. Hyperplane asymptotic conjecture. There exists a hyperplane $W\subset V_1$ such that

$$\lim\limits_{t\to\pm\infty}d(\pi\circ\gamma(t),W)=0.$$

This would sharpen the blowdown theorem by asserting that the horizontal projection of every such geodesic becomes asymptotic to a hyperplane in the first layer, rather than merely satisfying the weaker conclusion currently available. The source presents this as a possible improvement, and no resolution is given.