Fix a smooth projective variety $ X$ over the complex numbers.

We write $ H^n(X,\mathbf{Z}(d)) = \text{CH}^d(X, 2d-n)$ for Bloch’s higher Chow groups.

There is a cycle class map to Deligne cohomology $ $ c_{n,d}: H^n(X,\mathbf{Z}(d)) \to H^n_{\mathcal{D}}(X,\mathbf{Z}(d)).$ $

One can endow the Deligne cohomology groups with the structure of complex analytic spaces $ H^n_{\mathcal{D}}(X,d)$ in a canonical way. Write $ \text{Chow}_{d, 2d-n}(X)$ for the Chow variety of cycles on $ X$ of codimension $ d$ and degree $ 2d-n$ .

Can one realize $ c_{n,d}$ as induced by/factoring through a morphism $ \text{Chow}_{d,2d-n}(X)^{\rm an}\to H^n_{\mathcal{D}}(X,d)$ in any way?

I’m looking for something along the lines of point (2) before Remark 7 in these notes, for arbitrary codimension and degree https://static1.squarespace.com/static/57bf2a6de3df281593b7f57d/t/57bf63e33e00be08153d8c0a/1472160739280/homotopicalcycleclasses.pdf

I will benefit from any references.