  L. Capogna and M. Cowling showed that if $\phi$ is 1-quasiconformal on an
open subset of a Carnot group G, then composition with $\phi$ preserves
Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they
combine this with a regularity theorem for Q-harmonic functions to show that
$\phi$ is in fact $C^\infty$. As an application, they observe that a Liouville
type theorem holds for some Carnot groups of step 2.
  In this article we argue, using the Engel group as an example, that a
Liouville type theorem can be proved for every Carnot group. Indeed, the fact
that 1-quasiconformal maps are smooth allows us to obtain a Liouville type
theorem by applying the Tanaka prolongation theory.
