Desingularization conjecture for orbifold Euler characteristics of quotients

Let $G$ be a finite group, or more generally a linear algebraic group, acting on a variety $X$, and let $X\to Y:=X/G$ be the quotient. A $G$-variety is a variety with a $G$-action, and a proper $G$-morphism is a proper morphism compatible with the $G$-actions. Let $\tilde{X}$ be a $G$-variety and let $f:\tilde{X}\to X$ be a proper $G$-morphism. The conjecture concerns the equivariant pushforward $f^G_*$, the equivariant Chern class transformation $C^G_*$, and the constructible functions $1\hskip-3.5pt1_{\tilde{X}/G, quo}$ and $1\hskip-3.5pt1_{X/G, orb}$ associated with the quotient and orbifold theories. Desingularization conjecture for orbifold Euler characteristics. There exists such a $G$-variety $\tilde{X}$ and proper $G$-morphism $f$ satisfying

$$f^G_*(1\hskip-3.5pt1_{\tilde{X}/G, quo})=1\hskip-3.5pt1_{X/G, orb}.$$

Consequently,

$$f^G_*C^G_*(1\hskip-3.5pt1_{\tilde{X}/G, quo})=C^G_*(1\hskip-3.5pt1_{X/G, orb}),$$

and the zeroth degrees satisfy

$$\chi(\tilde{X}/G)=\chi(X;G).$$

The conjecture proposes that the orbifold invariants of a quotient can be realized through a suitable equivariant desingularization. The source gives no resolution status or further conditions on the construction of $\tilde{X}$ and $f$, so the claim is recorded as open.