Are rays in Carnot groups straight?

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 $G \simeq \mathfrak{g}$).