Are rays in Carnot groups straight? A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also this question). This is open also in the special case of Carnot groups.
When $G$ is a Carnot group, one can try at least to prove that, given a geodesic $\gamma:[0,T]\to G$ with $\gamma(0)=e$, the family of geodesics
$$ \gamma_r:[0,r^{-1}T]\to G,\quad\gamma_r(t):=\delta_{r^{-1}}(\gamma(rt)) $$
tend locally uniformly to a straight line, as $r\to 0$. (It is well-known that Carnot groups come equipped with a one-parameter family of automorphisms $(\delta_r)_{r\in\mathbb{R}^+}$ satisfying $d(\delta_r(x),\delta_r(y))=rd(x,y)$. Here $d$ is the Carnot-Carathéodory distance.)
It is easy to see that any limit curve $\gamma_\infty:[0,\infty)\to G$ along a subsequence of $r$'s is a ray, i.e. a length minimizer between any two of its points.
Hence a very natural question is the following:

Given any unit-speed ray $\alpha:[0,\infty)\to G$ (i.e. a continuous curve
  such that $d(\alpha(s),\alpha(t))=t-s$), is it true that $\alpha$ belongs to a one-parameter subgroup of $G$? In other words, is it true that $\alpha(t)=\exp(tX)$, for a suitable $X$ in the Lie algebra?

 A: This is far from being a complete answer, but it's too long for a comment.
Your statement is true, by direct inspection, for corank 1 and 2 Carnot groups, where the cut time of normal geodesics (or, equivalently, the cotangent injectivity domain for the normal exponential map) is known explicitly.
I am not sure whether the claim is true in general. However, if a counter-example exists, one should prove global optimality for a horizontal curve $\gamma : [0,+\infty)$ that is not a one-parameter subgroup. Even assuming that $\gamma$ is a normal geodesic (and thus solves some Hamiltonian equations) to simplify the task, this would be extremely hard to do (except for one-parameter subgroups, that coincide with straight lines in the first layer $\mathfrak{g}_1$ of the Carnot group $G \simeq \mathfrak{g}$).
A: No, infinite rays are not necessarily one-parameter subgroups.
An example can be found in the paper "Cut time in sub-riemannian problem on engel group" by Ardentov and Sachkov.
In the paper the geodesics of the Engel group (a rank 2, step 3 Carnot group) are separated into 7 different classes $C_1,...C_7$, of which $C_3, C_4, C_5$ and $C_7$ give infinite geodesics. The geodesics of class $C_3$ give examples of infinite rays that are not one-parameter subgroups.
