The comparison conjecture for Hopf-cyclic cohomology of algebras with coefficients

Let $A$ be an algebra, $(R,I)$ a coefficient pair, ${\mathcal H}$ a Hopf algebra, and ${\mathcal M}$ coefficients as in the construction of the complexes ${\mathcal X}_A$ and ${\mathcal X}_{\mathcal H}(R,I;{\mathcal M})$. The map of complexes in Proposition induces a map on homology theories.

Comparison conjecture. There is an isomorphism of homology theories

$$HC^*(A,R,I;{\mathcal H},{\mathcal M})\cong HC^*({\mathcal X}_A,{\mathcal X}_{\mathcal H}(R,I;{\mathcal M})).$$

This conjecture asserts that the Hopf-cyclic homology theory with coefficients agrees with the homology theory of the associated supercomplexes. The source provides no resolution or evidence of a proof, so the conjecture is left open.