The inverse-limit comparison conjecture for Hopf-cyclic cohomology of coalgebras

Let $C$ be a coalgebra, ${\mathcal M}$ a coefficient module, and ${\mathcal H}$ a Hopf algebra. Let $W(C)$ and $I(C)$ be the constructions appearing in the associated supercomplex ${\mathcal X}_{\mathcal H}(W(C),I(C);{\mathcal M})$. The map

$$HC^*_{\mathcal H}(C,{\mathcal M})\to H^*({\mathcal X}_{\mathcal H}(W(C),I(C);{\mathcal M}))$$

is defined by the inverse-limit construction described before the conjecture.

Inverse-limit comparison conjecture. The map above is an isomorphism of the cohomology groups.

This conjecture proposes that Hopf-cyclic cohomology of the coalgebra with coefficients is recovered by the cohomology of the associated supercomplex. The supplied text gives no resolution or status evidence, so it remains open.