Sp(n)U(1)Sp(n)U(1)-connections with parallel totally skew-symmetric torsion

We consider the unique Hermitian connection with totally skew-symmetric torsion on a Hermitian manifold. We prove that if the torsion is parallel and the holonomy is Sp(n)U(1)⊂U(2n)×U(1)Sp(n)U(1)⊂U(2n)×U(1), then the manifold is locally isomorphic to the twistor space of a quaternionic Kähler manifold with positive scalar curvature. If the manifold is complete, then it is globally isomorphic to such a twistor space.