Topological T-duality for Real and equivariant circle bundles

Let $X$ be a space with a $\mathbb{Z}/2$-action. A T-dual pair consists of a pair $(E,h)$, where $\mathbb{Z}/2$ acts equivariantly on the circle bundle $pi:E\to X$ and $h\in H^3_\pm(E)$, and a pair $(\hat E,\hat h)$, where $\hat E\to X$ is a Real circle bundle and $\hat h\in H^3_\pm(\hat E)$. Write $p$ and $\hat p$ for the maps from a common correspondence space to $E$ and $\hat E$, respectively, and let $c_1^{\mathbb{Z}/2}(E)\in H^2_{\mathbb{Z}/2}(X)$ denote the equivariant Chern class of $E$. Topological T-duality conjecture. For every such $X$, every pair $(E,h)$ has a pair $(\hat E,\hat h)$ satisfying

$$c_1^R(\hat E)=\pi_*h,\qquad c_1^{\mathbb{Z}/2}(E)=\hat\pi_*\hat h,\qquad c_1^{\mathbb{Z}/2}(E)\cup c_1^R(\hat E)=0,$$

and

$$p^*h=\hat p^*\hat h.$$

Conversely, every pair $(\hat E,\hat h)$ has such a pair $(E,h)$, and the two assignments induce mutually inverse maps on isomorphism classes of pairs. This proposes a duality interchanging Real and equivariant circle bundles while relating their twisted cohomological data; the paper presents it as a possible topological T-duality rather than an established theorem.