The Grothendieck ring of a non-divisible ordered abelian group is trivial

We consider the model-theoretic Grothendieck ring of definable sets in ordered abelian groups. It is well-known that $\mathrm{K}\mathbb{Q}\cong \mathbb{Z}[T]\mathrm{/}(T^2+T)$ and $\mathrm{K}\mathbb{Z}=0$, but surprisingly little is known about other cases. We present a short computation which shows that they all collapse: $\mathrm{K}G=0$, unless $G$ is divisible.