Generalized Carlson–Clemens–Morgan conjecture for crossover mappings

Let $X$ be a complex algebraic manifold and let $d$ be a positive integer. Write $QH^k(X)$ for the quotient cohomology groups appearing in the exact sequence associated with iterated integrals, and let $\bar\break\Phi$ be the homomorphism constructed from the extension of mixed Hodge structures and the Chow-group crossover map. For a quasi-projective complex algebraic manifold $X$, the generalized Carlson–Clemens–Morgan conjecture. The mapping $\bar\Phi$ equals the composition of the crossover mapping with the quotient mapping

$$J^d(X)\to J(QH^{2d+1}(X)(d)).$$

If true, this would generalize the theorem of Carlson, Clemens and Morgan from simply connected projective manifolds and codimension-one cycles to quasi-projective manifolds and cycles of all codimensions; the source presents it as unproved.