Almost continuous orbit equivalence for non-singular homeomorphisms

Let $X$ and $Y$ be Polish spaces with non-atomic Borel measures $\mu$ and $\nu$ of full support. Suppose that $T$ and $S$ are ergodic non-singular homeomorphisms of $(X,\mu)$ and $(Y,\nu)$ with continuous Radon-Nikodym derivatives. Suppose that either they are both of type $III_1$ or that they are both of type $III_\lambda$, $0<\lambda<1$ and, in the $III_\lambda$ case, suppose in addition that both `topological asymptotic ranges' (defined in the article) are $\log\lambda\cdot\Bbb Z$. Then there exist invariant dense $G_\delta$-subsets $X'\subset X$ and $Y'\subset Y$ of full measure and a non-singular homeomorphism $\phi: X' \to Y'$ which is an orbit equivalence between $T|_{X'}$ and $S|_{Y'}$, that is $\phi\{T^{i}x\} = \{S^{i}x\}$ for all $x \in X'$. Moreover the Radon-Nikodym derivative $d\nu\circ\phi/d\mu$ is continuous on $X'$ and, letting $S' = \phi^{-1}S \phi$ we have $Tx= {S'}^{n(x)}x$ and $S' = T^{m(x)}x$ where $n$ and $m$ are continuous on $X'$.