Quotient singularities in the Grothendieck ring of varieties

Let $G$ be a finite group, $X$ be a smooth complex projective variety with a faithful $G$-action, and $Y$ be a resolution of singularities of $X\mathrm{/}G$. Larsen and Lunts asked whether $[X\mathrm{/}G]-[Y]$ is divisible by $[{\mathbb{A}}^1]$ in the Grothendieck ring of varieties. We show that the answer is negative if $BG$ is not stably rational and affirmative if $G$ is abelian. The case when $X=Z^n$ for some smooth projective variety $Z$ and $G=S_n$ acts by permutation of the factors is of particular interest. We make progress on it by showing that $[Z^n\mathrm{/}{S}_n]-[Z⟨n⟩\mathrm{/}{S}_n]$ is divisible by $[{\mathbb{A}}^1]$, where $Z⟨n⟩$ is Ulyanov’s polydiagonal compactification of the $n$-th configuration space of $Z$.