Identity conjecture for the unitary-conjugation descent construction

Let $A_\theta$ be the irrational rotation algebra, let $\mathcal{G}_{A_\theta}$ be its unitary conjugation groupoid, and let $u\in M_n(A_\theta)$ be invertible. Consider the proposed maps from the class of $u$ through equivariant $KK$-theory, groupoid descent, and the Morita identification:

$$[u]\in K_1(A_\theta)\xrightarrow{\mathrm{lift}}KK^1_{\mathcal{G}_{A_\theta}}(A_\theta,\mathbb C)\xrightarrow{\mathrm{desc}}K_1(C^*(\mathcal{G}_{A_\theta}))\xrightarrow{\cong}K_1(A_\theta).$$

Identity conjecture for the unitary-conjugation descent construction. The resulting class coincides with the original class $[u]$. In particular, the lift map is injective and its image consists exactly of those equivariant $KK$-classes fixed by the natural automorphisms.

The text describes this as a fundamental conjecture planned for future work, intended to give an explicit geometric realization of $K_1(A_\theta)$ through the unitary conjugation groupoid.