Let $MGL$ denote the motivic spectrum representing algebraic cobordism. Over $\mathop{Spec}(\mathbb{C})$ there is a Betti realization functor $\mathop{SH}(\mathbb{R}) \to \mathop{SH}$, which takes $MGL$ to $MU$.

Over $\mathop{Spec}(\mathbb{R})$ there is a $C_2$-equivariant Betti realization functor $\mathop{SH}(\mathbb{R}) \to \mathop{SH}(C_2)$. The analog of $MU$ in the $C_2$-equivariant homotopy category is $M\mathbb{R}$, the real bordism spectrum of Araki, Landweber, and Hu-Kriz, and it seems logical that the equivariant Betti realization takes $MGL$ to $M\mathbb{R}.$

Is there a reference that proves that the $C_2$-equivariant Betti realization of $MGL$ is $M\mathbb{R}$?

I have seen it claimed that this is proved in work of Hu and Kriz, but I was unable to find it there (that is not to say that it I didn't miss something!).

Note that it is known (for example, by work of Heller and Ormsby) Betti realization commutes with colimits. Since $MGL$ is a colimit of certain Thom spectra, one can reduce to checking the equivariant Betti realization of these.