The generalized projective Euler characteristic conjecture for finite groups

Let $d\geq 1$. For a finite group $G$, write ${\rm Cl}({\bf Z}[G])$ for the class group of the integral group ring. Let $Y$ be projective and flat of relative dimension $d$, let $X\to Y$ be a $G$-torsor, and let ${\mathcal F}$ be a $G$-equivariant coherent sheaf on $X$. Denote by $\bar\chi^P({\mathcal F})$ its projective Euler characteristic in ${\rm Cl}({\bf Z}[G])$.

Generalized projective Euler characteristic conjecture. There exists an integer $Q(d)$ such that, whenever $G$ is finite and ${\rm \gcd}(\#G,Q(d))=1$, one has

$$\bar\chi^P({\mathcal F})=0\quad\text{in }{\rm Cl}({\bf Z}[G])$$

for every such $G$-torsor and every $G$-equivariant coherent sheaf ${\mathcal F}$ on $X$. This extends the paper's vanishing results beyond the abelian or restricted Sylow-subgroup cases and remains open in the stated generality.