  Let $N\subset GL(n,R)$ be the group of upper triangular matrices with $1$s on
the diagonal, equipped with the standard Carnot group structure. We show that
quasiconformal homeomorphisms between open subsets of $N$, and more generally
Sobolev mappings with nondegenerate Pansu differential, are rigid when $n \geq
4$; this settles the Regularity Conjecture for such groups. This result is
deduced from a rigidity theorem for the manifold of complete flags in $R^n$.
Similar results also hold in the complex and quaternion cases.
