Let $X$ be a smooth projective complex analytic space, $i,j\ge 0$ integers, $\mathbf{Z}(j)_{\mathcal{D}}$ the Deligne complex of $X$, $H^i_{\mathcal{D}}(X,\mathbf{Z}(j))$ its hypercohomology.

What properties does the subgroup of torsion elements of $H^i_{\mathcal{D}}(X,\mathbf{Z}(j)$ have? 

For instance, is it finite? Does it contain divisible elements?

Since $H^i_{\mathcal{D}}(X,\mathbf{Z}(j))$ is an extension of a finitely generated $\mathbf{Z}$-module by a quotient of a graded in the Hodge filtration on de Rham cohomology of $X$, one expects the answer depends on $i,j$.

After a quick web search, it seems this is not in any standard reference in the literature.