Conjecture on cyclic homology of the crossed product by $\mathbb Z_2$

Let $\theta\notin\mathbb Q$, let $\mathcal A_\theta$ be the algebraic irrational rotational algebra, and let $\mathcal A_\theta\rtimes\mathbb Z_2$ denote its crossed product by the finite subgroup $\mathbb Z_2\subset SL(2,\mathbb Z)$. Write $HC_{even}$ and $HC_{odd}$ for its even and odd cyclic homology groups. Cyclic homology conjecture.

$$HC_{even}(\mathcal A_\theta\rtimes\mathbb Z_2)\cong\mathbb C^6,\qquad HC_{odd}(\mathcal A_\theta\rtimes\mathbb Z_2)=0.$$

This predicts the cyclic homology of the crossed product for every irrational parameter. The supplied text does not state whether the conjecture has been resolved.