  In this paper we formulate some conjectures in sub-Riemannian geometry
concerning a characterisation of the Koranyi-Kaplan ball in a group of
Heisenberg type through the existence of a solution to suitably overdetermined
problems. We prove an integral identity that provides a rigidity constraint for
one of the two problems. By exploiting some new invariances of these Lie
groups, for domains having partial symmetry we solve these problems by
converting them to known results for the classical $p$-Laplacian
