Nearly continuous Kakutani equivalence of adding machines

One says that two ergodic systems $(X,\mathcal F,\mu)$ and $(Y,\mathcal G,\nu)$ preserving a probability measure are evenly Kakutani equivalent if there exists an orbit equivalence $\phi: X\to Y$ such that, restricted to some subset $A\subseteq X$ of positive measure, $\phi$ becomes a conjugacy between the two induced maps $T_A$ and $S_{\phi(A)}$. It follows from the general theory of loosely Bernoulli systems developed in [8] that all adding machines are evenly Kakutani equivalent, as they are rank-1 systems. Recent work has shown that, in systems that are both topological and measure-preserving, it is natural to seek to strengthen purely measurable results to be "nearly continuous''. In the case of even Kakutani equivalence, what one asks is that the map $\phi$ and its inverse should be continuous on $G_\delta$ subsets of full measure and that the set $A$ should be within measure zero of being open and of being closed. What we will show here is that any two adding machines are indeed equivalent in this nearly continuous sense.