Cyclic cohomology conjecture for commutative extended Hopf algebras

Let $(H,R)$ be a commutative extended Hopf algebra, meaning that both $H$ and $R$ are commutative algebras. Denote by $H^i(H,R)$ the Hochschild cohomology of the cocyclic module $H_\natural$, with differential beginning with $d_0=\alpha-\beta$.

Cyclic cohomology conjecture. The cyclic cohomology of $(H,R)$ satisfies

$$HC^n(H)\cong \bigoplus_{i\geq 0} H^{n-2i}(H,R).$$

This conjecture extends the corresponding description for commutative Hopf algebras, where cyclic cohomology decomposes as a direct sum of Hochschild cohomology groups. The stated analogue is motivated by the validity of the related propositions for extended Hopf algebras; its resolution is not indicated here.