If there are nontrivial maps $H\mathbb{Z}^{top}\to MU$ then there are nontrivial maps $H\mathbb{Z}\to MGL$?

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$ ?

Are there non trivial maps from $H\mathbb{Z}$ to $MGL$?

Let $ k$ be a field of characteristic $ 0$ . Let us denote by $ \mathbf{1}_{k}$ the sphere spectrum. Let $ MGL$ be the algebraic cobordism spectrum.

We have the following diagram

$ $ H\mathbb{Z}\leftarrow \mathbf{1}_{k}\rightarrow MGL$ $

My question is the following:

Are there non trivial maps $ H\mathbb{Z}\to MGL$ such that the above triangle commutes?