Reid's homological McKay correspondence conjecture

Let $V$ be a vector space, let $G\subset SL(V)$ be a finite subgroup, and set $X=V/G$. Assume that there is a proper smooth crepant resolution $\pi:Y\to X$. For each homomorphism $g:\mu_r\to G$, let $v_g/r_g$ be the associated monomial valuation of $X$, let $\delta(v_g/r_g)\in Y$ be its center, and let $Z_g\subset Y$ be the Zariski closure of that center. Write $H^c_{\bullet}(Y,\mathbb{Q})$ for Borel–Moore homology and $\operatorname{cl}(Z_g)$ for the fundamental class of $Z_g$. Reid's conjecture. The classes $\operatorname{cl}(Z_g)$ of the algebraic cycles $Z_g$ form a basis of the $\mathbb{Q}$-vector space $H^c_{\bullet}(Y,\mathbb{Q})$. Moreover, for every $g:\mu_r\to G$,

$$\operatorname{age}(g)=\operatorname{codim}(Z_g).$$

This is the precise homological and cohomological McKay correspondence proposed by Reid. The supplied text says that the authors will prove it, so the claim is solved in this paper.