Parametrised T-duality trivialisation conjecture for the bulk-boundary correspondence

Let $X$ be the parameter space, let $H_1$, $H_2$, and $H_3$ be the twisting components on $X\times \mathbb T^n$, and write $Y_{H_2}$ for the principal $\mathbb T^n$-bundle over $X$ with Chern class $H_2$, with projection $q:Y_{H_2}\to X$ and parametrised deformation $\sigma$ determined by $H_1$. Choose a subgroup decomposition $\mathbb T^n=\mathbb T^{n-1}_a\times\mathbb T_b$, let $\iota:X\times\mathbb T^{n-1}_a\to X\times\mathbb T^n$ be the inclusion, and let $Y_{\iota^*H_2}$, $q_a$, and $\iota^*\sigma$ denote the corresponding restricted data. The bulk-boundary map is the Pimsner–Voiculescu boundary map

$$\partial:K_{j+n}\left({CT}(Y_{H_2},q^*(H_3))_\sigma\right)\longrightarrow K_{j+n-1}\left({CT}(Y_{\iota^*H_2},q_a^*(H_3))_{\iota^*\sigma}\right).$$

Parametrised T-duality trivialisation conjecture. The diagram

\xymatrix{ K^j(X\times \mathbb T^n,H_1+H_2+H_3) \ar[d]^{\iota^*} \ar[rr]^{\sim}_{T} && K_{j+n}\left({CT}(Y_{H_2},q^*(H_3))_\sigma\right) \ar[d]^\partial \\ K^j(X\times \mathbb T^{n-1},\iota^*H_1+\iota^*H_2+H_3) \ar[rr]^{\sim}_{T_a} && K_{j+n-1}\left({CT}(Y_{\iota^*H_2},q_a^*(H_3))_{\iota^*\sigma}\right) }

commutes, where $T_a$ is noncommutative T-duality with respect to $\mathbb T^{n-1}_a$ and $\iota^*$ is the induced restriction map in twisted $K$-theory. Thus T-duality trivialises the bulk-boundary correspondence in this parametrised context. This is posed as a general expectation for the construction described; no resolution is supplied in the source.