The weight-one rational-function conjecture for quotient singularities with crepant resolutions

Let $V$ be a vector space and let $G\subset SL(V)$ be finite. Set $X=V/G$, and assume that $X$ admits a smooth crepant resolution $Y\to X$. For a cyclic subgroup $g(\mu_r)\subset G$, let $g:\mu_r\to G$ denote the corresponding homomorphism, let $K(V)$ be the field of rational functions on $V$, and let $\chi$ be the fundamental character of $\mu_r$. Weight-one rational-function conjecture. For every cyclic subgroup $g(\mu_r)\subset G$, there exists a $G$-invariant rational function $f\in K(V)$ of weight $1$ for the induced $g(\mu_r)$-action; equivalently,

$$g(a)\cdot f=\chi(a)f$$

for every $a\in\mu_r$. This is an algebraic formulation of the preceding conjectural condition on cyclic subgroups. The paper relates it to the existence of smooth crepant resolutions, but the supplied text does not establish it in general.