Is there always an extremal teichmuller mapping?

Given a quasisymmetric self-homeomorphism h of the unit circle S-1, let Q(h) be the set of all quasiconformal mappings with the boundary correspondence h. In [1], it was shown that there exists h for which no extremal extension in Q(h) as a Teichmuller mapping is possible. This disproved some conjectures of long standing. In the example constructed there, the boundary correspondence has a single extremal quasiconformal extension. We show that even when there are infinitely many extremal extensions of the boundary values, it may still happen that none of the extensions is a Teichmuller mapping. An infinitesimal version of this result is also obtained.