The blow-up formulation of higher Todd genus birational invariance

Let $\phi\colon\widehat V\to V$ be a blowing down between nonsingular complex projective varieties, with $\widehat V$ obtained by blowing up $V$ along a nonsingular irreducible center of complex codimension at least $2$. Let $\pi$ be a discrete group, let $x\in H^*(B\pi,\mathbb Q)$, and let $u\colon V\to B\pi$ be a map. Blow-up invariance conjecture. Then

$$\langle \mathcal T(V)\cup u^*(x),[V]\rangle=\langle \mathcal T(\widehat V)\cup \phi^*\circ u^*(x),[\widehat V]\rangle.$$

This is the local blow-up form used to establish the birational-invariance conjecture via factorization of birational maps. The supplied text presents it as the reduction needed for the proof and gives no independent resolution status.