Let $ H\mathbb{Z}^{top}$ be the Eilenberg-MacLane spectrum and $ MU$ be the complex cobordism spectrum.

Consider now the motivic counterparts of the above spectral, namely $ MGL$ (the motivic cobordism) and $ H\mathbb{Z}$ (the motivic Eilenberg Mac-Lane spectra).

There is an adjunction $ (Re_{B},c):SH(\mathbb{C})\to SH$ , where $ Re_{B}$ is the Betti realization functor and $ c$ is the functor induced by sending a space to the constant presheaf of spaces on smooth varieties over $ \mathbb{C}$ . It is known that the functor $ c$ is fully faithful.

My question is the following:

If there are nontrivial maps $ H\mathbb{Z}^{top}\to MU$ , can I deduce that there are nontrivial maps $ H\mathbb{Z}\to MGL$ ?