Suslin's conjecture on morphic cohomology and singular cohomology

Let $\varepsilon\colon \mathop{\mathrm{Top}}\to\mathop{\mathrm{qProj}}_{\mathbb C}$ be the morphism of sites from the usual topology to the Zariski topology, and let $\mathbf{R}\varepsilon_*\mathbb Z$ be the derived push-forward of the constant sheaf $\mathbb Z$. For morphic cohomology, there is a natural comparison morphism

$$\mathcal{M}^*(q)\longrightarrow \mathbf{R}\varepsilon_*\mathbb Z.$$

Suslin's conjecture. The comparison morphism above is a quasi-isomorphism on smooth varieties.

This is the morphic analogue of the Beilinson–Lichtenbaum conjecture in the motivic setting. The supplied source does not state whether the conjecture has been resolved.