T-duality isomorphism for twisted Real K-theory

Let $(E,h)$ and $(\hat E,\hat h)$ be T-dual pairs of a $\mathbb{Z}/2$-equivariant circle bundle and a Real circle bundle, with twists $h\in H^3_\pm(E)$ and $\hat h\in H^3_\pm(\hat E)$. Denote by $KR^{h+n}(E)$ the Real K-theory of $E$ twisted by $h$. T-duality conjecture for twisted Real K-theory. For every integer $n$, there should be an isomorphism

$$KR^{h+n}(E)\cong KR^{\hat h+n-1}(\hat E).$$

This is the proposed K-theoretic counterpart of the preceding topological T-duality, predicting a degree shift by one between the twisted Real K-theories of T-dual pairs. The paper presents it as a proposal, and no resolution is supplied in the given text.