Factorization of the orientation map $MU\to H\mathbb{Z}$ through $ku$?

Question

Let $MU$ denote the complex cobordism spectrum and $ku$ the connective cover of the complex $K$-theory spectrum.

Is it true that the orientation map $MU\to H\mathbb{Z}$ factors through $ku$?

Answer 1

Yes, I think so. The Conner-Floyd map $MU \to ku$ is an isomorphism on $\pi_0$, and the maps $MU \to H\mathbb{Z}$ and $ku \to H\mathbb{Z}$ are just zeroth Postnikov truncations, which is functorial.