Conjecture on the cyclic cohomology of the smooth dihedral group-algebra subalgebra

Let $A^{\infty}$ be the smooth subalgebra of the group $C^*$-algebra associated with the dihedral group, and let $HC^n(A^{\infty})$ denote its cyclic cohomology in degree $n$. Cyclic cohomology conjecture. The cyclic cohomology of $A^{\infty}$ is given by

$$HC^{n}( A^{\infty} ) = \left\{ \begin{array}{cc} \mathbf{C}^3 &: n\,\,even, n \geq 2\\ 0 &: n\,\,odd, n \geq 1 \\ \end{array} \right.$$

This conjecture concerns whether the smooth subalgebra has any additional cyclic cohomology classes beyond those detected on the group ring; in particular, it asserts that there are no nonzero ghost cocycles in the indicated degrees. The surrounding discussion notes that the known degree-zero cocycles extend to $A^{\infty}$, but does not establish the full cyclic cohomology calculation.