Birational invariance of higher Todd genera

Let $M$ be a smooth projective variety, let $\pi$ be a group, and let $f:M\to B\pi$ be a continuous map. For $\alpha\in H^*(B\pi;\mathbb{Q})$, let $\operatorname{Td}(M)$ denote the Todd class of $M$ and let $[M]$ be its fundamental class. Birational invariance of higher Todd genera. The number

$$\left\langle f^*(\alpha)\cup \operatorname{Td}(M),[M]\right\rangle$$

is a birational invariant. The paper notes that the topological fundamental group is a birational invariant and says that Rosenberg established the claim for many groups; it then states that resolution of singularities and the Baum–Fulton–MacPherson Riemann–Roch theorem prove it in general.